Problem 14 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} y=x-2 \\ y=-3 x+2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the point of intersection, (1, -1).

Step 1 - Write Each Equation in Slope-Intercept Form

Both equations are already in the slope-intercept form, which is \( y = mx + b \). The first equation is \( y = x - 2 \) and the second equation is \( y = -3x + 2 \).

Step 2 - Identify the Slope and Y-intercept of Each Line

For the equation \( y = x - 2 \), the slope \( m \) is 1 and the y-intercept \( b \) is -2. For the equation \( y = -3x + 2 \), the slope \( m \) is -3 and the y-intercept \( b \) is 2.

Step 3 - Draw the First Line on the Graph

Start by plotting the y-intercept of the first equation \( y = x - 2 \), which is -2. Then use the slope (rise over run), which is 1, to plot another point. For every 1 unit you move up, move 1 unit to the right. Draw the line through these points.

Step 4 - Draw the Second Line on the Graph

Now, plot the y-intercept of the second equation \( y = -3x + 2 \), which is 2. Use the slope, which is -3, to plot another point. For every 3 units you move down, move 1 unit to the right. Draw the line through these points.

Step 5 - Find the Point of Intersection

Look at the graph to find the point where the two lines intersect. This point represents the solution to the system of equations.

Step 6 - Verify the Solution

Check the coordinates of the intersection point to ensure they satisfy both equations. Substitute these coordinates back into each original equation.

Key Concepts

slope-intercept form

When solving a system of linear equations by graphing, it's essential to understand the slope-intercept form of a linear equation. The slope-intercept form is written as: \( y = mx + b \). Here, \( m \) represents the slope of the line, which indicates how steep the line is, and \( b \) represents the y-intercept, where the line crosses the y-axis.
The slope-intercept form helps in quickly identifying key characteristics of the line, which will be used for graphing. In the given exercise, both equations \( y = x - 2 \) and \( y = -3x + 2 \) are already provided in slope-intercept form, making it easier to identify their slopes and y-intercepts.

graphical solution

A graphical solution involves solving a system of equations by plotting them on a graph to see where they intersect. Here’s how to find the solution graphically:

• First, rewrite each equation in the slope-intercept form if they aren't already.
• Next, identify the slope and y-intercept for each equation.
• Then, plot the y-intercept of each equation on the graph.
• Use the slope to identify additional points for each line.
• Draw the lines through these points.
• Finally, find the point where the two lines intersect.

The intersection point represents the solution to the system of equations. This is because it is the point where both equations are true simultaneously.

point of intersection

The point of intersection is crucial when solving systems of linear equations by graphing. It is where the two lines cross each other on the graph. This point represents the values of \( x \) and \( y \) that satisfy both equations.
In the provided exercise, after graphing the equations \( y = x - 2 \) and \( y = -3x + 2 \), the lines intersect at the point \( (1, -1) \). This implies that x=1 and y=-1 is the solution to the system. To confirm this, substitute these values back into the original equations to ensure both are satisfied.

linear equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They can be represented graphically as straight lines. Each linear equation can be expressed in slope-intercept form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
The two linear equations given in the exercise are \( y = x - 2 \) and \( y = -3x + 2 \). These equations graph into straight lines and provide a basis for understanding how different slopes and intercepts can affect where and how the lines intersect. Recognizing the characteristics of linear equations is fundamental in graphically solving systems of equations.