To graph the system of linear equations, we need to graph each equation separately on the same coordinate plane. Here are the equations:
1) \( y = x + 2 \)
2) \( y = -2x - 1 \)
(a) Graphing the system:
Step 1: Graph the first equation, \( y = x + 2 \).
- Start by finding the y-intercept which occurs when \( x = 0 \). In this case, \( y = 0 + 2 \), so the y-intercept is \( (0, 2) \).
- Next, determine the slope of the line. The equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the first equation, the slope (m) is 1.
- Plot the y-intercept on the graph.
- Use the slope to find another point. With a slope of 1, you rise 1 unit up and run 1 unit to the right from the y-intercept. This gives you the point \( (1, 3) \).
- Draw a straight line through these two points.
Step 2: Graph the second equation, \( y = -2x - 1 \).
- Start by finding the y-intercept which occurs when \( x = 0 \). In this case, \( y = -2(0) - 1 \), so the y-intercept is \( (0, -1) \).
- The slope of the line is -2. This means that for every 1 unit you move to the right on the x-axis, you move 2 units down on the y-axis.
- Plot the y-intercept on the same graph.
- Use the slope to find another point. Move 1 unit to the right from the y-intercept, then move 2 units down to find the point \( (1, -3) \).
- Draw a straight line through these two points.
Now you have two lines on the graph. The intersection point of these lines is the solution to the system of equations.
To find the intersection point by hand, we can set the two equations equal to each other since they both equal \( y \):
\( x + 2 = -2x - 1 \)
Solve for \( x \):
\( x + 2x = -1 - 2 \)
\( 3x = -3 \)
\( x = -1 \)
Now substitute \( x \) back into one of the original equations to find \( y \):
\( y = (-1) + 2 \)
\( y = 1 \)
Therefore, the solution to the system of equations, which is the intersection point of the two lines, is \( (-1, 1) \). On the graph, the two lines would intersect at this point.
