Problem 20 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} -3 x+y=-2 \\ 4 x-2 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (1, 1).

Step 1 - Rewrite Equations in Slope-Intercept Form

Rewrite each equation in the form of \( y = mx + b \). For the first equation, \begin{align*} -3x + y &= -2 \ \ y &= 3x - 2 done. For the second equation, \begin{align*} 4x - 2y &= 6 \ \ -2y &= -4x + 6 \ \ y &= 2x - 3done.This will make it easier to graph.

Step 2 - Graph the First Equation

Plot the equation \( y = 3x - 2 \). Start by plotting the y-intercept (0, -2). Then, use the slope (3) to plot another point. From (0, -2), move up 3 units and right 1 unit to plot the next point. Draw the line through these points.

Step 3 - Graph the Second Equation

Plot the equation \( y = 2x - 3 \). Start by plotting the y-intercept (0, -3). Then, use the slope (2) to plot another point. From (0, -3), move up 2 units and right 1 unit to plot the next point. Draw the line through these points.

Step 4 - Find the Point of Intersection

The points where the two lines intersect is the solution to the system of equations. The lines \( y = 3x - 2 \) and \( y = 2x - 3 \) intersect at the point (1, 1).

Step 5 - Verify the Solution

Verify the solution by substituting \( x = 1 \) and \( y = 1 \) back into the original equations. \begin{align*} -3(1) + 1 &= -2 \ \text{True} \ 4(1) - 2(1) &= 6 \ \text{True}. The point (1, 1) satisfies both equations, confirming it is the solution.

Key Concepts

Graphing

Graphing is a visual method used to solve systems of linear equations by plotting each equation on a coordinate plane.
To do this, you need to graph each line and identify where they intersect.
This intersection point represents the solution to the system.
For instance, graphing the equations from the example, we first converted them into slope-intercept form and then plotted them.
This helps to see visually where the two lines meet, making it easier to find the solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For example, given the original equations:
\begin{align*}-3x + y &= -2 \ 4x - 2y &= 6\text\by converting them, we get:
\begin{align*} y &= 3x - 2 \ y &= 2x - 3.
These forms are much easier to graph, emphasizing the slope of the line and the y-intercept.

Intersection Point

The intersection point of two lines on a graph is where the lines cross each other.
This point represents the solution to the system of equations.
It means that both equations are satisfied at this point.
In our example, the lines cross at the point (1, 1).
This means \( x = 1 \) and \( y = 1 \) is the solution to both equations.
Locating this point helps to find the values of the variables which satisfy both equations simultaneously.

Verification of Solution

Verification of the solution involves substituting the intersection point back into the original equations to ensure it satisfies both.
From the example, substituting \( (1, 1) \) back:

• For the first equation: \begin{align*}-3(1) + 1 &= -2 \ \text{True}
• For the second equation:\begin{align*}4(1) - 2(1)&= 6 \ \text{True}

Since both are true, the point (1, 1) is confirmed as the correct solution.
Verification ensures that the solution is accurate and reliable.