Solving Systems Of Equations A Step By Step Guide

JU07/13/2025 08, 2025 by THE IDEN 50 views

Solving Systems of Equations: A Comprehensive Guide

In mathematics, solving systems of equations is a fundamental skill with applications across various fields, from engineering and physics to economics and computer science. 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 satisfy all the equations simultaneously. In this article, we will delve into the process of solving a specific system of equations, providing a step-by-step guide and exploring the underlying concepts.

The System of Equations

We are presented with the following system of equations:

y = x^2 - 3x + 12
y = -2x + 14

This system consists of two equations. The first equation, y = x^2 - 3x + 12, is a quadratic equation representing a parabola. The second equation, y = -2x + 14, is a linear equation representing a straight line. To solve this system, we need to find the points where the parabola and the line intersect. These points of intersection represent the solutions to the system of equations, as they satisfy both equations simultaneously.

Methods for Solving Systems of Equations

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

Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation.
Elimination: This method involves adding or subtracting the equations to eliminate one variable.
Graphing: This method involves graphing the equations and finding the points of intersection.

For this particular system, the substitution method is the most straightforward approach. Since both equations are already solved for y, we can simply set the expressions for y equal to each other.

Step-by-Step Solution Using Substitution

Set the expressions for y equal to each other:
```
x^2 - 3x + 12 = -2x + 14
```
This step combines the two equations into a single equation with only one variable, x. By equating the two expressions for y, we are essentially finding the x-values where the parabola and the line have the same y-value, which corresponds to the points of intersection.
Rearrange the equation into a quadratic equation:
```
x^2 - 3x + 2x + 12 - 14 = 0
x^2 - x - 2 = 0
```
This step involves moving all the terms to one side of the equation, resulting in a standard quadratic equation in the form ax^2 + bx + c = 0. This form is essential for applying various methods to solve for x, such as factoring, completing the square, or using the quadratic formula.
Solve the quadratic equation:

The quadratic equation can be solved by factoring. We are looking for two numbers that multiply to -2 and add to -1. These numbers are -2 and 1.
```
(x - 2)(x + 1) = 0
```
This step involves factoring the quadratic expression into two binomial factors. The factors represent the values of x that make the expression equal to zero. Setting each factor equal to zero allows us to solve for the individual x-values.
Set each factor equal to zero and solve for x:
```
x - 2 = 0  or  x + 1 = 0
x = 2  or  x = -1
```
This step yields the two possible x-values that satisfy the quadratic equation. These x-values represent the x-coordinates of the points of intersection between the parabola and the line.
Substitute the values of x back into either of the original equations to find the corresponding y-values:

Let's use the simpler equation, y = -2x + 14.
- For x = -1:
```
y = -2(-1) + 14
y = 2 + 14
y = 16
```
  This substitution gives us the y-coordinate corresponding to x = -1. By plugging the x-value into the linear equation, we find the y-value on the line at that particular x-coordinate.
- For x = 2:
```
y = -2(2) + 14
y = -4 + 14
y = 10
```
  Similarly, this substitution gives us the y-coordinate corresponding to x = 2. This calculation determines the y-value on the line when x is equal to 2.
Write the solutions as ordered pairs (x, y):

The solutions to the system of equations are (-1, 16) and (2, 10).

These ordered pairs represent the points where the parabola and the line intersect on the coordinate plane. They are the solutions because they satisfy both equations simultaneously.

Verification of Solutions

To ensure the accuracy of our solutions, we can substitute the x and y values back into both original equations and verify that they hold true.

For the solution (-1, 16):

Equation 1: y = x^2 - 3x + 12

16 = (-1)^2 - 3(-1) + 12
16 = 1 + 3 + 12
16 = 16 (True)

Equation 2: y = -2x + 14

16 = -2(-1) + 14
16 = 2 + 14
16 = 16 (True)

For the solution (2, 10):

Equation 1: y = x^2 - 3x + 12

10 = (2)^2 - 3(2) + 12
10 = 4 - 6 + 12
10 = 10 (True)

Equation 2: y = -2x + 14

10 = -2(2) + 14
10 = -4 + 14
10 = 10 (True)

Since both solutions satisfy both equations, we can confidently conclude that they are the correct solutions to the system of equations.

Conclusion

In this article, we have successfully solved a system of equations involving a quadratic equation and a linear equation. We employed the substitution method, which proved to be an efficient approach for this particular system. By setting the expressions for y equal to each other, we reduced the system to a single quadratic equation, which we then solved by factoring. The resulting x-values were substituted back into one of the original equations to find the corresponding y-values. Finally, we verified our solutions by plugging them back into both original equations.

Solving systems of equations is a crucial skill in mathematics and its applications. Understanding the different methods available and choosing the most appropriate one for a given system is essential. With practice and a solid grasp of the underlying concepts, you can confidently tackle a wide range of systems of equations.

The solutions to the system of equations are (-1, 16) and (2, 10).

Therefore, the completed statements are:

One solution is (-1, 16). The second solution is (2, 10).