Solving System Of Linear Equations 2x + 3y = 3 And 7x - 3y = 24

This article delves into the process of solving a system of linear equations, using the specific example provided:

$\begin{array}{l} 2 x+3 y=3 \\ 7 x-3 y=24 \end{array}$

We will explore various methods to arrive at the solution, including the elimination method, and provide a detailed explanation of each step. Understanding how to solve systems of linear equations is a fundamental skill in algebra and has wide-ranging applications in various fields, including engineering, economics, and computer science. Linear equations, at their core, represent relationships between variables where the graph of the equation forms a straight line. When we have a system of two or more linear equations, we are essentially looking for the point(s) where these lines intersect. This intersection point represents the solution that satisfies all equations in the system simultaneously. In real-world scenarios, systems of linear equations can model diverse situations, such as determining the break-even point for a business, optimizing resource allocation, or analyzing the flow of traffic in a network. The ability to solve these systems accurately is crucial for making informed decisions and solving complex problems. The elimination method is a powerful technique that allows us to eliminate one variable from the system by adding or subtracting the equations. This simplifies the system, making it easier to solve for the remaining variable. We then substitute the value of this variable back into one of the original equations to find the value of the other variable. This process provides a systematic way to find the solution that satisfies all equations in the system. In this article, we will break down the elimination method step-by-step, illustrating its application with the given example. By mastering this method, you will gain a valuable tool for solving a wide range of linear equation systems and tackling related problems effectively. So, let's embark on this journey to unravel the intricacies of solving systems of linear equations and equip ourselves with the knowledge to conquer these mathematical challenges.

Understanding the Problem

Before diving into the solution, let's clearly state the problem. We are given a system of two linear equations:

$\begin{array}{l} 2 x+3 y=3 \\ 7 x-3 y=24 \end{array}$

Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously. This means finding the point of intersection of the two lines represented by these equations. The solution will be an ordered pair $(x, y)$ that, when substituted into both equations, makes both equations true. Understanding the problem clearly is the first crucial step in solving any mathematical problem. In this case, we recognize that we are dealing with a system of two linear equations, each representing a straight line on a graph. The solution we seek is the point where these two lines intersect, if such a point exists. It is important to note that not all systems of linear equations have a unique solution. Some systems may have no solution, indicating that the lines are parallel and never intersect. Other systems may have infinitely many solutions, meaning that the two equations represent the same line. However, in this particular problem, we are given a system that is likely to have a unique solution, and our task is to find that solution. To achieve this, we will employ algebraic techniques to manipulate the equations and isolate the variables. This process will lead us to the values of $x$ and $y$ that satisfy both equations, thus giving us the coordinates of the intersection point. By carefully analyzing the equations and applying the appropriate methods, we can confidently arrive at the solution. So, let's proceed with the next step, where we will choose a method to solve the system and begin the process of finding the values of $x$ and $y$ that satisfy both equations.

Method 1: Elimination Method

The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated. Notice that in our system, the coefficients of $y$ are $3$ and $-3$. This makes the elimination method a particularly good choice for this problem. The elimination method is a powerful technique for solving systems of linear equations. Its core principle lies in manipulating the equations in such a way that, when added or subtracted, one of the variables is effectively canceled out. This simplification allows us to solve for the remaining variable, and subsequently, find the value of the eliminated variable. In the given system of equations, we observe a favorable situation: the coefficients of the $y$ variable are $3$ and $-3$. This provides an immediate opportunity to eliminate $y$ by simply adding the two equations together. The beauty of the elimination method lies in its efficiency and clarity. By strategically manipulating the equations, we can avoid complex substitutions or rearrangements, leading to a straightforward solution. This method is particularly advantageous when the coefficients of one of the variables are opposites or multiples of each other, as in our case. By adding the equations, we will create a new equation with only one variable, making it easy to solve. Once we find the value of that variable, we can substitute it back into one of the original equations to determine the value of the other variable. This step-by-step process ensures that we arrive at the solution that satisfies both equations simultaneously. So, let's proceed with the elimination method and witness how it elegantly solves the system of equations, guiding us to the values of $x$ and $y$ that make both equations true. By mastering this method, you will enhance your problem-solving skills and gain confidence in tackling various mathematical challenges involving linear equations.

Step 1: Add the Equations

Adding the two equations, we get:

$(2x + 3y) + (7x - 3y) = 3 + 24$

This simplifies to:

$9x = 27$

The first step in the elimination method is to carefully add the two equations together. This seemingly simple operation is the key to eliminating one of the variables and simplifying the system. By adding the equations, we combine the corresponding terms on both sides of the equations. In our case, we have $(2x + 3y) + (7x - 3y)$ on the left-hand side and $3 + 24$ on the right-hand side. The magic of this step lies in the fact that the $y$ terms, $3y$ and $-3y$, cancel each other out. This is because they are additive inverses, meaning they have the same magnitude but opposite signs. When added together, they sum up to zero, effectively eliminating the $y$ variable from the equation. This elimination is the core principle of the elimination method, allowing us to reduce the system to a single equation with a single variable. After adding the equations and simplifying, we arrive at the equation $9x = 27$. This equation is much easier to solve than the original system of two equations. It directly relates the variable $x$ to a constant, allowing us to isolate $x$ and find its value. The simplicity of this equation is a testament to the power of the elimination method. By strategically adding the equations, we have transformed a complex system into a simple equation that can be readily solved. So, let's proceed to the next step, where we will isolate $x$ and find its value, bringing us closer to the solution of the system.

Step 2: Solve for $x$

Dividing both sides of the equation $9x = 27$ by $9$, we get:

$x = 3$

Now that we have eliminated $y$ and obtained the simplified equation $9x = 27$, the next logical step is to solve for the variable $x$. This involves isolating $x$ on one side of the equation, which will reveal its value. To isolate $x$, we need to undo the multiplication by $9$. The inverse operation of multiplication is division, so we divide both sides of the equation by $9$. This maintains the equality of the equation while effectively separating $x$ from its coefficient. Dividing both sides of $9x = 27$ by $9$ yields $x = 3$. This result tells us that the $x$-coordinate of the solution to the system of equations is $3$. This is a significant step forward in solving the system, as we have now determined the value of one of the variables. With the value of $x$ in hand, we can proceed to find the value of $y$. This will involve substituting the value of $x$ back into one of the original equations and solving for $y$. The process of solving for $x$ demonstrates the power of algebraic manipulation. By applying the appropriate operations to both sides of the equation, we can isolate the variable of interest and determine its value. This skill is fundamental in solving a wide range of mathematical problems and is a cornerstone of algebraic thinking. So, let's move on to the next step, where we will substitute $x = 3$ into one of the original equations and solve for $y$, completing the process of finding the solution to the system.

Step 3: Substitute $x$ into one of the original equations to solve for $y$

Let's use the first equation:

$2(3) + 3y = 3$

Simplifying, we get:

$6 + 3y = 3$

Subtracting 6 from both sides:

$3y = -3$

Dividing by 3:

$y = -1$

Having found the value of $x$, the next crucial step is to determine the value of $y$. This is achieved by substituting the value of $x$ into one of the original equations. The choice of which equation to use is arbitrary, as both equations will lead to the same value of $y$. However, it is often prudent to choose the equation that appears simpler or easier to work with. In this case, let's opt for the first equation, $2x + 3y = 3$. Substituting $x = 3$ into this equation, we get $2(3) + 3y = 3$. This substitution replaces the variable $x$ with its numerical value, transforming the equation into one with only $y$ as the unknown. This is a pivotal moment in the solution process, as we are now poised to isolate $y$ and find its value. Simplifying the equation, we have $6 + 3y = 3$. To isolate $y$, we first subtract $6$ from both sides of the equation, resulting in $3y = -3$. This step moves the constant term to the right-hand side, bringing us closer to isolating $y$. Finally, we divide both sides of the equation by $3$ to solve for $y$. This yields $y = -1$. We have now successfully found the value of $y$, which is $-1$. This completes the process of solving the system of equations. We have determined that $x = 3$ and $y = -1$. These values, when substituted into both original equations, will satisfy both equations simultaneously. Therefore, the solution to the system is the ordered pair $(3, -1)$. So, let's summarize our findings and present the solution clearly.

Step 4: Write the solution as an ordered pair

The solution to the system of equations is $(3, -1)$. This ordered pair represents the point where the two lines intersect on a graph. Expressing the solution as an ordered pair is the final step in solving the system of equations. The ordered pair $(3, -1)$ represents a specific point on the coordinate plane. The first number in the pair, $3$, is the $x$-coordinate, and the second number, $-1$, is the $y$-coordinate. This point is the unique solution to the system, meaning that it satisfies both equations simultaneously. When we graph the two linear equations, the lines will intersect at this point. To verify that $(3, -1)$ is indeed the solution, we can substitute these values back into the original equations and check if the equations hold true. For the first equation, $2x + 3y = 3$, substituting $x = 3$ and $y = -1$ gives $2(3) + 3(-1) = 6 - 3 = 3$, which is true. For the second equation, $7x - 3y = 24$, substituting $x = 3$ and $y = -1$ gives $7(3) - 3(-1) = 21 + 3 = 24$, which is also true. Since the ordered pair $(3, -1)$ satisfies both equations, we can confidently conclude that it is the correct solution. The ordered pair representation provides a concise and clear way to express the solution to a system of linear equations. It encapsulates the values of both variables that satisfy the system, making it easy to interpret and use. So, we have successfully solved the system of equations and expressed the solution as an ordered pair. This completes the process and provides a definitive answer to the problem.

Method 2: Substitution Method (Alternative Approach)

While the elimination method is efficient for this particular system, the substitution method provides another valid approach. Let's demonstrate how it works. The substitution method is an alternative approach to solving systems of linear equations. Unlike the elimination method, which aims to eliminate one variable by adding or subtracting equations, the substitution method focuses on expressing one variable in terms of the other and substituting this expression into the other equation. This method is particularly useful when one of the equations is already solved for one variable or can be easily rearranged to solve for one variable. The core idea behind the substitution method is to reduce the system of two equations with two variables into a single equation with a single variable. This is achieved by expressing one variable as a function of the other and then replacing that variable in the other equation with its expression. This process eliminates one variable, leaving us with an equation that can be solved for the remaining variable. Once we find the value of that variable, we can substitute it back into one of the original equations or the expression we derived earlier to find the value of the other variable. The substitution method is a versatile technique that can be applied to a wide range of systems of linear equations. It is especially effective when one equation is already in a form that makes substitution straightforward. By mastering this method, you will expand your problem-solving toolkit and gain the ability to tackle systems of equations from different perspectives. So, let's explore how the substitution method can be applied to the given system of equations and see how it leads us to the same solution we obtained using the elimination method. By understanding both methods, you will be well-equipped to choose the most efficient approach for any given system of equations.

Step 1: Solve one equation for one variable

Let's solve the first equation for $x$:

$2x + 3y = 3$

Subtract $3y$ from both sides:

$2x = 3 - 3y$

Divide by 2:

$x = \frac{3 - 3y}{2}$

The first step in the substitution method is to choose one of the equations and solve it for one of the variables. This means isolating one variable on one side of the equation, expressing it in terms of the other variable. The choice of which equation and which variable to solve for is often guided by convenience. We look for an equation where one of the variables has a coefficient of $1$ or $-1$, as this will minimize the complexity of the algebraic manipulations. In our system of equations, neither variable has a coefficient of $1$ or $-1$. However, let's choose the first equation, $2x + 3y = 3$, and solve it for $x$. This choice is arbitrary, and we could have chosen to solve for $y$ or used the second equation instead. To solve for $x$, we first subtract $3y$ from both sides of the equation, resulting in $2x = 3 - 3y$. This step isolates the term containing $x$ on the left-hand side. Next, we divide both sides of the equation by $2$ to completely isolate $x$. This gives us the expression $x = \frac{3 - 3y}{2}$. We have now expressed $x$ in terms of $y$. This expression tells us how the value of $x$ depends on the value of $y$. It is a crucial component of the substitution method, as we will use it to substitute for $x$ in the other equation. This step demonstrates the power of algebraic manipulation in rearranging equations to suit our needs. By carefully applying the rules of algebra, we have transformed the equation into a form that allows us to express one variable in terms of the other. So, let's proceed to the next step, where we will substitute this expression for $x$ into the second equation.

Step 2: Substitute this expression into the other equation

Substitute $x = \frac{3 - 3y}{2}$ into the second equation:

$7(\frac{3 - 3y}{2}) - 3y = 24$

Now that we have expressed $x$ in terms of $y$, the next step in the substitution method is to substitute this expression into the other equation. This is the core of the substitution method, as it allows us to eliminate one variable and obtain an equation with only one variable. We substitute the expression $x = \frac{3 - 3y}{2}$ into the second equation, $7x - 3y = 24$. This gives us the equation $7(\frac{3 - 3y}{2}) - 3y = 24$. This equation looks more complex than the original equations, but it is a significant step forward because it contains only one variable, $y$. By substituting for $x$, we have effectively eliminated $x$ from the equation. The resulting equation is a linear equation in $y$, which we can solve using standard algebraic techniques. The substitution process can sometimes lead to equations with fractions or other complexities, but it is a powerful tool for solving systems of equations. By carefully performing the substitution and simplifying the resulting equation, we can isolate the remaining variable and find its value. This step highlights the interconnectedness of the variables in a system of equations. By expressing one variable in terms of the other and substituting, we reveal the relationship between the variables and pave the way for finding their values. So, let's proceed to the next step, where we will simplify this equation and solve for $y$, bringing us closer to the solution of the system.

Step 3: Solve for $y$

Multiplying both sides by 2 to eliminate the fraction:

$7(3 - 3y) - 6y = 48$

Expanding:

$21 - 21y - 6y = 48$

Combining like terms:

$21 - 27y = 48$

Subtracting 21 from both sides:

$-27y = 27$

Dividing by -27:

$y = -1$

After substituting for $x$ in the second equation, we obtained the equation $7(\frac{3 - 3y}{2}) - 3y = 24$. The next step is to solve this equation for $y$. This involves a series of algebraic manipulations to isolate $y$ on one side of the equation. First, to eliminate the fraction, we multiply both sides of the equation by $2$. This gives us $7(3 - 3y) - 6y = 48$. Next, we expand the expression on the left-hand side by distributing the $7$, resulting in $21 - 21y - 6y = 48$. Now, we combine the like terms involving $y$, which gives us $21 - 27y = 48$. To isolate the term containing $y$, we subtract $21$ from both sides of the equation, resulting in $-27y = 27$. Finally, we divide both sides of the equation by $-27$ to solve for $y$. This gives us $y = -1$. We have now successfully found the value of $y$ using the substitution method. This value agrees with the value we found using the elimination method, which is a good indication that our calculations are correct. The process of solving for $y$ demonstrates the importance of careful and methodical algebraic manipulation. By following the order of operations and applying the rules of algebra correctly, we can isolate the variable of interest and determine its value. So, let's proceed to the next step, where we will substitute this value of $y$ back into one of the equations to solve for $x$.

Step 4: Substitute $y$ back into the expression for $x$

Substitute $y = -1$ into $x = \frac{3 - 3y}{2}$:

$x = \frac{3 - 3(-1)}{2}$

Simplifying:

$x = \frac{3 + 3}{2}$

$x = \frac{6}{2}$

$x = 3$

Having found the value of $y$, the final step in the substitution method is to substitute this value back into the expression we derived earlier for $x$. This will allow us to determine the value of $x$ and complete the solution to the system of equations. We substitute $y = -1$ into the expression $x = \frac{3 - 3y}{2}$. This gives us $x = \frac{3 - 3(-1)}{2}$. Now, we simplify the expression by following the order of operations. First, we multiply $-3$ by $-1$, which gives us $3$. So, the expression becomes $x = \frac{3 + 3}{2}$. Next, we add $3$ and $3$ in the numerator, which gives us $6$. So, the expression becomes $x = \frac{6}{2}$. Finally, we divide $6$ by $2$, which gives us $x = 3$. We have now found the value of $x$, which is $3$. This value agrees with the value we found using the elimination method. This confirms that our solution is consistent and correct. The process of substituting the value of $y$ back into the expression for $x$ highlights the interconnectedness of the variables in the system. By using the relationship we established earlier, we can easily find the value of the remaining variable. So, we have now found the values of both $x$ and $y$. Let's proceed to the final step, where we will write the solution as an ordered pair.

Step 5: Write the solution as an ordered pair

The solution to the system of equations is $(3, -1)$, which matches the result we obtained using the elimination method.

The solution to the system of equations is the ordered pair $(3, -1)$. This ordered pair represents the point where the two lines represented by the equations intersect on the coordinate plane. The $x$-coordinate of the point is $3$, and the $y$-coordinate is $-1$. This means that when we substitute $x = 3$ and $y = -1$ into both of the original equations, the equations will hold true. We have now successfully solved the system of equations using both the elimination method and the substitution method. Both methods led us to the same solution, which provides strong evidence that our solution is correct. Expressing the solution as an ordered pair is a standard way of representing the solution to a system of two linear equations in two variables. It provides a concise and clear way to communicate the values of the variables that satisfy the system. The ordered pair $(3, -1)$ represents a specific point on the coordinate plane, and it is the unique solution to the system of equations. This means that there is no other pair of values for $x$ and $y$ that will satisfy both equations simultaneously. So, we have confidently arrived at the solution to the system of equations. This completes the process and provides a definitive answer to the problem. By mastering both the elimination method and the substitution method, you have equipped yourself with powerful tools for solving a wide range of systems of linear equations.

Conclusion

The solution to the system of linear equations

$\begin{array}{l} 2 x+3 y=3 \\ 7 x-3 y=24 \end{array}$

is (C) $(3, -1)$. We have demonstrated two methods, elimination and substitution, to arrive at this solution. In conclusion, we have successfully solved the given system of linear equations using two distinct methods: the elimination method and the substitution method. Both methods led us to the same solution, which is the ordered pair $(3, -1)$. This ordered pair represents the point of intersection of the two lines represented by the equations. The elimination method involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This simplifies the system, allowing us to solve for the remaining variable. We then substitute the value of this variable back into one of the original equations to find the value of the other variable. The substitution method, on the other hand, involves solving one equation for one variable and then substituting this expression into the other equation. This eliminates one variable, allowing us to solve for the remaining variable. We then substitute the value of this variable back into the expression we derived earlier to find the value of the other variable. The fact that both methods led us to the same solution reinforces the correctness of our answer. It also demonstrates the versatility of algebraic techniques in solving systems of equations. Understanding both methods provides a comprehensive approach to solving these types of problems and allows you to choose the method that is most efficient for a given system. So, we have confidently concluded that the solution to the system of linear equations is $(3, -1)$. This completes the process and provides a definitive answer to the problem. By mastering these methods, you will be well-equipped to tackle a wide range of problems involving systems of linear equations.