Problem 49 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \( (2, 2) \).

Step 1: Convert the equations to slope-intercept form

Ensure both equations are in the form of \( y = mx + b \). The second equation \( y = -x + 4 \) is already in this form. For the first equation, solve for \( y \):Starting with \( -3x + 2y = -2 \):Add \( 3x \) to both sides: \( 2y = 3x - 2 \)Divide by 2: \( y = \frac{3}{2}x - 1 \).

Step 2: Identify the slope and y-intercept of both equations

For the first equation \( y = \frac{3}{2}x - 1 \), the slope \( m \) is \( \frac{3}{2} \) and the y-intercept \( b \) is -1.For the second equation \( y = -x + 4 \), the slope \( m \) is -1 and the y-intercept \( b \) is 4.

Step 3: Plot the y-intercepts on the graph

Plot \( (0, -1) \) for the first equation and \( (0, 4) \) for the second equation on the coordinate plane.

Step 4: Use the slopes to plot additional points

For the first equation \( y = \frac{3}{2}x - 1 \), start at \( (0, -1) \) and use the slope \( \frac{3}{2} \) to move up 3 units and right 2 units to plot the next point \( (2, 2) \).For the second equation \( y = -x + 4 \), start at \( (0, 4) \) and use the slope -1 to move down 1 unit and right 1 unit to plot the next point \( (1, 3) \).

Step 5: Draw the lines for both equations

Use a ruler to draw a line through the points plotted for each equation. Extend the lines to see where they intersect.

Step 6: Identify the point of intersection

The point where the two lines intersect represents the solution to the system of equations. Based on the graph, the lines intersect at the point \( (2, 2) \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a way to visually represent mathematical relationships. It involves plotting points on a coordinate plane and drawing lines through these points. Each equation can be represented as a straight line.
The key steps include identifying the y-intercept and using the slope to determine additional points to plot. Once you have enough points, you can draw a line.
For example, with the given equations, you start plotting points for each one based on their specific formulas.

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 is practical for graphing because it quickly tells you where to start and how to move on the graph.
For instance, in the equation \( y = -x + 4 \), the slope is -1, and the y-intercept is 4. This means you begin at point (0, 4) and use the slope to determine the line's direction.
The slope \( m \) indicates the steepness of the line. The y-intercept \( b \) shows where the line crosses the y-axis. Understanding both elements makes plotting the graph straightforward.

Intersection Point

The intersection point is where two lines on a graph meet. This point represents the solution to a system of linear equations.
To find this point, graph each equation on the same coordinate plane. The coordinates of the intersection are the x and y values that satisfy both equations simultaneously.
In our example, the lines from \( y = \frac{3}{2}x - 1 \) and \( y = -x + 4 \) intersect at (2, 2).
This means that when \( x = 2 \), both equations are true, confirming \( x = 2 \) and \( y = 2 \) as the solution.