Solutions To The System Of Equations Y=x^2+3x-7 And 3x-y=-2

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In the realm of mathematics, solving systems of equations is a fundamental skill. A system of equations is a set of two or more equations that share 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. There are several methods for solving systems of equations, including substitution, elimination, and graphing. In this article, we will explore how to find the solutions to a specific system of equations using the substitution method. The given system involves a quadratic equation and a linear equation, which adds an interesting layer to the solution process. Understanding these methods is crucial for various applications in science, engineering, and economics, where mathematical models often involve multiple interconnected equations.

We are given the following system of equations:

y = x^2 + 3x - 7
3x - y = -2

Our objective is to determine the solutions for this system. This means we need to find the pairs of (x, y) values that satisfy both equations. The options provided are:

A. (3, 11) and (-3, -7) B. (11, 3) and (-3, -7) C. (3, 11) and (-7, -3) D. No real solutions

To solve this, we will employ the method of substitution, which involves solving one equation for one variable and substituting that expression into the other equation. This approach allows us to reduce the system to a single equation in one variable, which we can then solve using standard algebraic techniques. Let's delve into the step-by-step solution.

To solve the system of equations, we will use the substitution method. This involves the following steps:

  1. Isolate one variable in one of the equations: We can start by isolating y in the second equation. The second equation is 3x - y = -2. We can rewrite this equation to solve for y:

    y = 3x + 2

  2. Substitute the expression into the other equation: Now we substitute this expression for y into the first equation, which is y = x^2 + 3x - 7. Substituting y with 3x + 2 gives us:

    3x + 2 = x^2 + 3x - 7

  3. Simplify and solve for x: Now we have a quadratic equation in terms of x. Let's simplify it by moving all terms to one side:

    0 = x^2 + 3x - 7 - 3x - 2 0 = x^2 - 9

    This is a difference of squares, which can be factored as:

    0 = (x - 3)(x + 3)

    Setting each factor equal to zero gives us the solutions for x:

    x - 3 = 0 or x + 3 = 0 x = 3 or x = -3

  4. Substitute the values of x back to find the corresponding y values: Now we substitute each value of x back into the equation y = 3x + 2 to find the corresponding y values:

    For x = 3: y = 3(3) + 2 = 9 + 2 = 11

    For x = -3: y = 3(-3) + 2 = -9 + 2 = -7

  5. Write the solutions as ordered pairs: Thus, the solutions are the ordered pairs (3, 11) and (-3, -7).

To ensure our solutions are correct, we need to substitute the (x, y) pairs back into both original equations and verify that they satisfy both. This step is crucial to avoid errors and confirm the accuracy of our solution. Let's verify each solution:

  1. For the solution (3, 11):

    • First equation: y = x^2 + 3x - 7 Substituting x = 3 and y = 11: 11 = (3)^2 + 3(3) - 7 11 = 9 + 9 - 7 11 = 11 (The first equation is satisfied)
    • Second equation: 3x - y = -2 Substituting x = 3 and y = 11: 3(3) - 11 = -2 9 - 11 = -2 -2 = -2 (The second equation is satisfied)

  2. For the solution (-3, -7):

    • First equation: y = x^2 + 3x - 7 Substituting x = -3 and y = -7: -7 = (-3)^2 + 3(-3) - 7 -7 = 9 - 9 - 7 -7 = -7 (The first equation is satisfied)
    • Second equation: 3x - y = -2 Substituting x = -3 and y = -7: 3(-3) - (-7) = -2 -9 + 7 = -2 -2 = -2 (The second equation is satisfied)

Since both solutions satisfy both equations, we can confidently conclude that our solutions are correct.

Now that we have found the solutions, let's review the given options to select the correct one:

A. (3, 11) and (-3, -7) B. (11, 3) and (-3, -7) C. (3, 11) and (-7, -3) D. No real solutions

Our solutions are (3, 11) and (-3, -7). Comparing these with the options, we find that option A matches our solutions exactly.

The correct answer is A. (3, 11) and (-3, -7). These are the points where the parabola and the line intersect in the Cartesian plane. Graphically, this means that if we were to plot these two equations, they would intersect at the points (3, 11) and (-3, -7). This graphical interpretation provides a visual confirmation of our algebraic solution, enhancing our understanding of the relationship between the equations and their solutions. Understanding such graphical interpretations is crucial in higher mathematics, particularly in calculus and analytical geometry, where the visual representation of equations and their solutions plays a vital role.

In this article, we successfully found the solutions to the given system of equations using the substitution method. We isolated y in the linear equation, substituted it into the quadratic equation, and solved for x. We then substituted the x values back to find the corresponding y values. Finally, we verified the solutions by plugging them back into the original equations and confirmed that they satisfy both. This systematic approach ensures the accuracy of our results. Mastering the solution of systems of equations is essential for various mathematical and real-world applications. The ability to solve such systems opens doors to more advanced topics in mathematics and provides valuable tools for problem-solving in numerous fields. The process of verification not only confirms the correctness of the solution but also reinforces the understanding of the underlying mathematical principles.