Problem 24 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x+y=6 \\ x-y=-8 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (7, -1).

Step 1: Rewrite the equations in slope-intercept form

First, rewrite each equation in the form of y = mx + b.For the equation \( x + y = 6 \):\[ x + y = 6 \]Subtract x from both sides:\[ y = -x + 6 \].For the equation \( x - y = -8 \):\[ x - y = -8 \]Add y to both sides and then add 8 to both sides:\[ x = y - 8 \]Rewrite it as:\[ y = x + 8 \]

Step 2: Graph the first equation

Graph the equation \( y = -x + 6 \).Start with the y-intercept (0, 6) and use the slope -1 to plot another point. From (0, 6), move down 1 unit and right 1 unit to plot another point at (1, 5). Draw a line through these points.

Step 3: Graph the second equation

Graph the equation \( y = x + 8 \).Start with the y-intercept (0, 8) and use the slope 1 to plot another point. From (0, 8), move up 1 unit and right 1 unit to plot another point at (1, 9). Draw a line through these points.

Step 4: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. Observe the graph and find the intersection point.

Step 5: Verify the solution

Substitute the coordinates of the intersection point back into the original equations to verify they satisfy both equations. If the solution satisfies both, it is correct.

Key Concepts

system of linear equations

In algebra, a system of linear equations consists of two or more linear equations containing the same set of variables. Solving these systems means finding the values of the variables that satisfy all the equations simultaneously.
This particular exercise involves two equations:
\( \begin{array}{l} x + y = 6 ewline x - y = -8 \end{array} \).
There are several methods to solve these equations, such as substitution, elimination, and graphing. In this exercise, we'll focus on the graphing method. Graphing helps you visually find the point where the equations intersect. This intersection point represents the solution to the system.

slope-intercept form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form makes it easier to graph the equation.
In our exercise:

• For the first equation, \( x + y = 6 \), we subtract \( x \) from both sides to get \( y = -x + 6 \). Here, the slope \( m = -1 \), and the y-intercept \( b = 6 \).
• For the second equation, \( x - y = -8 \), we rewrite it to \( y = x + 8 \). The slope \( m = 1 \), and the y-intercept \( b = 8 \).

Rewriting the equations in slope-intercept form simplifies the graphing process.

graphing equations

Graphing linear equations involves plotting points and drawing lines based on the slope and y-intercept.
Here's how we graph the equations from our exercise:

• Graph the first equation, \( y = -x + 6 \):
  Start at the y-intercept (0, 6). Since the slope \( m = -1 \), move down 1 unit and right 1 unit to reach another point at (1, 5). Draw a line through these points.
• Graph the second equation, \( y = x + 8 \):
  Start at the y-intercept (0, 8). With slope \( m = 1 \), move up 1 unit and right 1 unit to plot another point at (1, 9). Draw a line through these points.

The point where the two lines intersect is the solution to the system. Verify this point by substituting it back into the original equations to ensure it satisfies both. This confirms the accuracy of your graphing.