Problem 10 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The lines intersect at (-2, 0), which is the solution.

Step 1: Rewrite each equation in slope-intercept form

Convert both equations into the form of y = mx + b, where m is the slope and b is the y-intercept.For the first equation -x + y = 2:Add x to both sides to isolate y:\( y = x + 2 \)For the second equation 2x + y = -4:Subtract 2x from both sides to isolate y:\( y = -2x - 4 \)

Step 2: Plot the y-intercepts

Identify and plot the y-intercepts (b) of both equations on a graph.The y-intercept of the first equation y = x + 2 is 2 (point (0,2)).The y-intercept of the second equation y = -2x - 4 is -4 (point (0, -4)).

Step 3: Use the slopes to plot additional points

Using the slopes (m) from each equation, plot additional points for each line.For the first equation y = x + 2, the slope is 1, meaning rise over run is 1/1.From (0, 2), go up 1 unit and right 1 unit to get to (1, 3).For the second equation y = -2x - 4, the slope is -2, meaning rise over run is -2/1.From (0, -4), go down 2 units and right 1 unit to get to (1, -6).

Step 4: Draw the lines

Draw straight lines through the points plotted for each equation.The first line passes through (0, 2) and (1, 3).The second line passes through (0, -4) and (1, -6).

Step 5: Identify the intersection point

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

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a way to express the equation so it's easy to understand and graph. The general form is:
\[y = mx + b\]

Here, 'm' represents the slope, and 'b' represents the y-intercept.
To convert an equation into slope-intercept form, you'll need to solve for 'y'. For example, if you start with the equation -x + y = 2, you add 'x' to both sides to get:
\[y = x + 2\]
Similarly, for the equation 2x + y = -4, you'd subtract 2x from both sides:
\[y = -2x - 4\]

This form makes it easier to identify the slope and y-intercept, which are essential for graphing the equations.

graphing linear equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them. Once an equation is in slope-intercept form, you can easily identify the y-intercept and the slope:
\[y = mx + b\]

Start by plotting the y-intercept on the graph. This is where the line crosses the y-axis.
For example, in the equation y = x + 2, the y-intercept is 2, so you plot a point at (0, 2).
Next, use the slope to find other points on the line. The slope tells you how much 'y' changes for a change in 'x'.

If the slope 'm' is 1, you move up 1 unit on the y-axis and right 1 unit on the x-axis.
Plot this new point. Repeat this process to find more points if needed.
Then, draw a straight line through the points.

intersection point

The intersection point of two lines on a graph is where the lines cross each other. This point represents the solution to a system of linear equations.
In other words, it is the set of coordinates (x, y) that satisfies both equations.

To find the intersection point, graph both equations on the same coordinate plane.
The point where the two lines meet is the solution.
For example, if you graph the equations y = x + 2 and y = -2x - 4, you will see that the lines intersect at a specific point.
By looking at the graph, you can identify the coordinates of this point.
In this exercise, the lines intersect at the point ( -2, 0). This means the solution to the system of equations is x = -2 and y = 0.