Solving The System Of Equations Y=-5x+3 And Y=1 A Step-by-Step Guide

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Solving systems of equations is a fundamental concept in algebra, and it's essential for various mathematical and real-world applications. In this article, we will delve into the process of solving the given system of equations:

y = -5x + 3
y = 1

We will explore the steps involved in finding the solution, which represents the point where these two equations intersect on a graph. By the end of this guide, you'll have a clear understanding of how to solve this specific system and similar problems.

Understanding Systems of Equations

Before diving into the solution, let's first understand what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. In graphical terms, the solution represents the point(s) where the lines or curves represented by the equations intersect.

In our case, we have two equations:

  1. y = -5x + 3
  2. y = 1

The first equation is a linear equation in slope-intercept form, where -5 is the slope and 3 is the y-intercept. The second equation is a horizontal line where the y-value is always 1.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, including:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
  • Graphing Method: This method involves graphing the equations and finding the point(s) of intersection.

For this particular system, the substitution method is the most straightforward approach.

Solving the System Using Substitution

Since we already have y isolated in both equations, the substitution method is the most efficient way to solve this system. Here's how it works:

  1. Substitute the value of y from the second equation into the first equation: Since y = 1, we can substitute 1 for y in the first equation:

    1 = -5x + 3

  2. Solve for x: Now we have an equation with only one variable, x. Let's solve for x:

    • Subtract 3 from both sides:
      1 - 3 = -5x + 3 - 3
-2 = -5x
    • Divide both sides by -5:
      -2 / -5 = -5x / -5
x = 2/5
x = 0.4

  3. Find the value of y: We already know that y = 1 from the second equation.

  4. Write the solution as an ordered pair: The solution is the ordered pair (x, y), which is (0.4, 1). This means the point where the two lines intersect is at x = 0.4 and y = 1.

Graphical Interpretation

To further understand the solution, let's consider the graphical interpretation. The equation y = -5x + 3 represents a line with a slope of -5 and a y-intercept of 3. The equation y = 1 represents a horizontal line passing through the point (0, 1).

If you were to graph these two lines, you would see that they intersect at the point (0.4, 1). This point is the solution to the system of equations because it satisfies both equations simultaneously.

Verification

To verify our solution, we can plug the values of x and y back into the original equations:

  1. Equation 1: y = -5x + 3
    1 = -5(0.4) + 3
1 = -2 + 3
1 = 1  (True)
  2. Equation 2: y = 1
    1 = 1  (True)

Since the values x = 0.4 and y = 1 satisfy both equations, we have confirmed that (0.4, 1) is indeed the correct solution.

Common Mistakes to Avoid

When solving systems of equations, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are a few to keep in mind:

  • Incorrect Substitution: Make sure to substitute the expression for the variable correctly. Double-check your work to avoid errors.
  • Arithmetic Errors: Pay close attention to arithmetic operations, especially when dealing with negative numbers or fractions.
  • Forgetting to Solve for Both Variables: Remember that the solution to a system of equations includes values for all the variables. Don't stop after solving for just one variable.
  • Misinterpreting the Solution: The solution is the point of intersection, not just the x or y value. Write the solution as an ordered pair (x, y).

Alternative Methods

While the substitution method was the most efficient for this particular system, let's briefly discuss other methods for solving systems of equations.

Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one variable. In this case, we could rewrite the second equation as y - 1 = 0 and then subtract it from the first equation. However, since we already have y isolated in both equations, the substitution method is more straightforward.

Graphing Method

The graphing method involves graphing the equations and finding the point(s) of intersection. While this method can be useful for visualizing the solution, it may not be as precise as algebraic methods, especially when the solution involves non-integer values. In our case, graphing the lines would visually confirm the intersection at (0.4, 1).

Real-World Applications

Systems of equations have numerous real-world applications in various fields, including:

  • Economics: Determining supply and demand equilibrium.
  • Engineering: Analyzing electrical circuits or structural systems.
  • Physics: Solving problems involving motion and forces.
  • Computer Graphics: Calculating intersections and transformations.
  • Finance: Modeling investments and loans.

Understanding how to solve systems of equations is a valuable skill that can be applied in many practical situations.

Conclusion

In this article, we have thoroughly explored the process of solving the system of equations:

y = -5x + 3
y = 1

We used the substitution method to find the solution, which is (0.4, 1). We also discussed the graphical interpretation of the solution, verified our answer, and highlighted common mistakes to avoid. Additionally, we briefly touched upon alternative methods and real-world applications of systems of equations.

By mastering the techniques for solving systems of equations, you'll be well-equipped to tackle a wide range of mathematical problems and real-world scenarios. Remember to practice regularly and apply these concepts to various situations to solidify your understanding.

Are you grappling with systems of equations? Do the lines and variables seem to blur into a confusing mess? Fear not! This comprehensive guide will walk you through the process of solving the system of equations:

y = -5x + 3
y = 1

We'll break down each step, ensuring you grasp the underlying concepts and emerge confident in your ability to tackle similar problems. Let's embark on this mathematical journey together!

What is a System of Equations?

Before diving into the solution, let's establish a solid foundation. A system of equations is simply a set of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Think of it as finding the common ground where all the equations agree.

In our case, we have two equations:

  1. y = -5x + 3 (a linear equation representing a line)
  2. y = 1 (another linear equation, specifically a horizontal line)

Our mission is to find the values of 'x' and 'y' that make both of these equations true. Graphically, this corresponds to finding the point where the two lines intersect.

Methods to Solve Systems of Equations

There are several approaches to solving systems of equations, each with its strengths and weaknesses. The most common methods include:

  • Substitution Method: This involves solving one equation for one variable and then substituting that expression into the other equation. This is often the most efficient method when one equation is already solved for a variable or can be easily manipulated.
  • Elimination Method: This method involves manipulating the equations so that when they are added or subtracted, one of the variables cancels out. This is particularly useful when the coefficients of one variable are opposites or multiples of each other.
  • Graphing Method: This involves plotting the equations on a graph and visually identifying the point(s) of intersection. While intuitive, this method may not be precise for solutions involving fractions or decimals.

For our specific system, the substitution method shines as the most direct and efficient path to the solution.

The Substitution Method: A Step-by-Step Walkthrough

The beauty of the substitution method lies in its simplicity. Since we already have 'y' isolated in both equations, the process becomes remarkably straightforward.

Step 1: Substitute the Value of 'y'

The second equation, y = 1, tells us that the value of 'y' is consistently 1. We can leverage this information by substituting '1' for 'y' in the first equation:

1 = -5x + 3

Notice how we've effectively eliminated 'y' from the first equation, leaving us with a single equation in terms of 'x'.

Step 2: Solve for 'x'

Now we have a simple algebraic equation to solve for 'x'. Let's isolate 'x' step-by-step:

  • Subtract 3 from both sides:
1 - 3 = -5x + 3 - 3
-2 = -5x
  • Divide both sides by -5:
-2 / -5 = -5x / -5
x = 2/5
  • Simplify the fraction (optional): x = 0.4

We've successfully determined the value of 'x'! x = 0.4

Step 3: Determine the Value of 'y'

This step is almost too easy! We already know that y = 1 from the second equation. No further calculations are needed.

Step 4: Express the Solution as an Ordered Pair

The solution to a system of equations is represented as an ordered pair (x, y). In our case, the solution is (0.4, 1). This means that the point where the two lines intersect on a graph is at x = 0.4 and y = 1.

Visualizing the Solution: The Graphical Perspective

To deepen our understanding, let's visualize the solution graphically. The equation y = -5x + 3 represents a line with a slope of -5 and a y-intercept of 3. The equation y = 1 represents a horizontal line that intersects the y-axis at 1.

If you were to plot these two lines on a graph, you would observe that they indeed intersect at the point (0.4, 1). This provides a visual confirmation of our algebraic solution.

Verification: Ensuring Accuracy

It's always prudent to verify our solution to ensure accuracy. We can do this by plugging the values of 'x' and 'y' back into the original equations.

Verification for Equation 1: y = -5x + 3

Substitute x = 0.4 and y = 1:

1 = -5(0.4) + 3
1 = -2 + 3
1 = 1  (True)

The equation holds true!

Verification for Equation 2: y = 1

This is straightforward: 1 = 1 (True)

Since the values x = 0.4 and y = 1 satisfy both equations, we can confidently declare that (0.4, 1) is the correct solution.

Common Pitfalls to Avoid

Solving systems of equations can sometimes be tricky, so let's highlight some common mistakes to watch out for:

  • Substitution Errors: Double-check that you're substituting the correct expression and that you're distributing any coefficients properly.
  • Arithmetic Mistakes: Pay close attention to signs and calculations, especially when dealing with negative numbers or fractions.
  • Incomplete Solutions: Remember that the solution must include values for all variables in the system. Don't stop after solving for just one variable.
  • Misinterpreting the Solution: The solution is the point of intersection, not just the individual x or y values. Express the solution as an ordered pair (x, y).

Alternative Solution Methods (Brief Overview)

While the substitution method was optimal for this system, let's briefly touch upon other methods.

Elimination Method

The elimination method involves manipulating equations to cancel out a variable. Although applicable here, it's less efficient than substitution since 'y' is already isolated in one equation.

Graphing Method

The graphing method offers a visual approach but may lack precision for non-integer solutions. In our case, plotting the lines would visually confirm the intersection at (0.4, 1).

Real-World Applications of Systems of Equations

Systems of equations aren't just abstract mathematical concepts; they have wide-ranging applications in various fields:

  • Economics: Modeling supply and demand curves to determine equilibrium prices.
  • Engineering: Analyzing circuits, structures, and systems with multiple interacting components.
  • Physics: Solving problems involving motion, forces, and energy conservation.
  • Computer Graphics: Calculating intersections, transformations, and rendering complex scenes.
  • Finance: Modeling investments, loans, and financial planning scenarios.

Conclusion: Mastering the Art of Solving Systems of Equations

In this comprehensive guide, we've meticulously dissected the process of solving the system of equations:

y = -5x + 3
y = 1

We employed the substitution method to arrive at the solution (0.4, 1). We also explored the graphical interpretation, verified our answer, identified common pitfalls, and touched upon alternative methods and real-world applications.

Armed with these techniques and insights, you're now well-equipped to confidently tackle a wide array of systems of equations. Remember, practice is key! The more you apply these concepts, the more fluent and adept you'll become at solving these problems.

Are you ready to unravel the mystery behind solving systems of equations? This comprehensive guide will walk you through a detailed, step-by-step process of finding the solution for the following system:

y = -5x + 3
y = 1

We'll break down each step with clarity, ensuring you not only find the answer but also understand the underlying principles. Buckle up, math enthusiasts; let's dive into the world of equations!

What Does It Mean to Solve a System of Equations?

Before we jump into the mechanics, it's crucial to grasp the fundamental concept. A system of equations is a collection of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all equations in the system true simultaneously. Imagine it as finding the perfect coordinates that fit every equation's requirement.

In our case, we have two equations:

  1. y = -5x + 3 (This is a linear equation, meaning it represents a straight line when graphed.)
  2. y = 1 (This is also a linear equation, representing a horizontal line.)

Our goal is to pinpoint the single (x, y) coordinate pair that satisfies both of these equations. Graphically, this solution corresponds to the point where the two lines intersect.

The Toolbox: Methods for Solving Systems of Equations

There are several established methods for tackling systems of equations, each with its own strengths and best-use scenarios. Let's briefly survey the landscape:

  • Substitution Method: This technique involves solving one equation for one variable and then substituting that expression into the other equation. It's particularly effective when one equation is already solved for a variable or can easily be manipulated.
  • Elimination Method: Also known as the addition or subtraction method, this approach aims to eliminate one variable by adding or subtracting the equations in a way that cancels out one of the terms. This works well when coefficients of one variable are opposites or multiples of each other.
  • Graphing Method: This intuitive method involves plotting the equations on a graph and visually identifying the intersection point(s). It's great for visualization but may not yield precise solutions for systems with fractional or decimal answers.

For our specific system, the substitution method stands out as the most efficient and elegant solution path.

Substitution in Action: A Step-by-Step Solution

The substitution method is our weapon of choice for this system, thanks to its straightforward application. Let's break it down:

Step 1: Leverage the Direct Substitution

The beauty of our system lies in the second equation, y = 1. It explicitly tells us the value of 'y'. We can immediately use this information and substitute '1' for 'y' in the first equation:

1 = -5x + 3

Notice how we've successfully replaced 'y' in the first equation, leaving us with a single equation in terms of 'x'.

Step 2: Isolate and Solve for 'x'

Now we're dealing with a simple algebraic equation. Our mission is to isolate 'x'. Let's proceed step-by-step:

  • Subtract 3 from both sides of the equation:
1 - 3 = -5x + 3 - 3
-2 = -5x
  • Divide both sides by -5:
-2 / -5 = -5x / -5
x = 2/5
  • Simplify the fraction (optional): x = 0.4

Success! We've found the value of 'x': x = 0.4

Step 3: The Value of 'y' is Already Known

This is where our system shines. The second equation, y = 1, has already gifted us with the value of 'y'. No further calculations are required.

Step 4: Express the Solution as an Ordered Pair

The solution to a system of equations is always presented as an ordered pair (x, y). This convention clearly indicates the values of both variables. In our case, the solution is (0.4, 1). This means the lines representing these equations intersect at the point where x = 0.4 and y = 1.

Seeing is Believing: The Graphical Interpretation

To solidify our understanding, let's visualize the solution. The equation y = -5x + 3 corresponds to a straight line with a slope of -5 and a y-intercept of 3. The equation y = 1 represents a horizontal line intersecting the y-axis at 1.

If you were to graph these two lines, you'd observe them intersecting precisely at the point (0.4, 1). This visual confirmation reinforces the accuracy of our algebraic solution.

Double-Checking Our Work: Verification

It's a hallmark of good mathematical practice to verify our results. Let's plug the values of 'x' and 'y' back into the original equations to ensure they hold true.

Verification for Equation 1: y = -5x + 3

Substitute x = 0.4 and y = 1:

1 = -5(0.4) + 3
1 = -2 + 3
1 = 1  (True)

The equation is satisfied!

Verification for Equation 2: y = 1

This is self-evident: 1 = 1 (True)

Since the values x = 0.4 and y = 1 satisfy both equations, we've definitively confirmed that (0.4, 1) is indeed the correct solution.

Avoiding Common Mishaps: Potential Pitfalls

Solving systems of equations can sometimes be a minefield of potential errors. Let's shed light on some common pitfalls to help you navigate these problems with confidence:

  • Careless Substitution: Ensure you're substituting the expression accurately and distributing coefficients correctly.
  • Arithmetic Errors: Meticulously check your calculations, particularly when dealing with negative numbers or fractions.
  • Incomplete Solutions: Remember to find the values of all variables in the system. Don't stop after solving for just one.
  • Misinterpreting the Solution: The solution is the intersection point, not just the individual x or y values. Always express your solution as an ordered pair (x, y).

A Glimpse at Other Methods (A Brief Overview)

While the substitution method was optimal for our system, let's briefly acknowledge alternative approaches.

Elimination Method

While applicable, the elimination method is less efficient here since 'y' is already isolated in one equation.

Graphing Method

Graphing provides a visual approach but may lack precision for non-integer solutions. In our case, it would visually confirm the intersection at (0.4, 1).

Systems of Equations in the Real World

Systems of equations aren't confined to textbooks; they're powerful tools for modeling real-world phenomena:

  • Economics: Determining equilibrium prices and quantities in supply and demand models.
  • Engineering: Analyzing electrical circuits, structural integrity, and complex systems.
  • Physics: Solving motion problems, force interactions, and energy balance.
  • Computer Graphics: Calculating intersections, transformations, and rendering 3D scenes.
  • Finance: Modeling investment portfolios, loan scenarios, and financial planning.

Conclusion: Unlocking the Power of Systems of Equations

In this detailed guide, we've meticulously solved the system of equations:

y = -5x + 3
y = 1

We harnessed the substitution method to arrive at the solution (0.4, 1). We also delved into the graphical interpretation, verified our answer, identified potential pitfalls, and explored alternative methods and real-world applications.

Equipped with these insights and techniques, you're now well-prepared to confidently solve a wide range of systems of equations. Remember, practice makes perfect! The more you work with these concepts, the more proficient you'll become in this essential area of mathematics.