To solve the system of equations graphically, we need to plot both equations on the same set of axes and find their intersection point.
1. Equation 1: [tex]\( y = x + 5 \)[/tex]
This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = 1 \)[/tex] (slope) and [tex]\( b = 5 \)[/tex] (y-intercept).
To plot this line:
- Start by plotting the y-intercept, which is at the point [tex]\( (0, 5) \)[/tex].
- Use the slope, which is [tex]\( 1 \)[/tex]. From [tex]\( (0, 5) \)[/tex], move 1 unit up and 1 unit to the right to plot another point. This gives the point [tex]\( (1, 6) \)[/tex].
- Draw a straight line through these points.
2. Equation 2: [tex]\( 2x + y = -7 \)[/tex]
We can rewrite this equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[
y = -2x - 7
\][/tex]
Here, the slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\(-7\)[/tex].
To plot this line:
- Start by plotting the y-intercept, which is at the point [tex]\( (0, -7) \)[/tex].
- Use the slope, which is [tex]\(-2\)[/tex]. From [tex]\( (0, -7) \)[/tex], move 2 units down and 1 unit to the right to plot another point. This gives the point [tex]\( (1, -9) \)[/tex].
- Draw a straight line through these points.
3. Finding the Intersection Point:
The intersection of these two lines represents the solution to the system of equations. The intersection point is where both equations are satisfied simultaneously.
From our graphs, the intersection point is found to be [tex]\( (-4, 1) \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
x = -4, \quad y = 1
\][/tex]
This means that both lines intersect at the point [tex]\((-4, 1)\)[/tex], which is the solution to the system of equations.
