Question
Solve the following system of equations graphically on the set of axes below. y, equals, one half, x, plus, 7 y= 2 1 x+7 y, equals, minus, 2, x, minus, 8 y=−2x−8
Answers
Answered by
GPT-4o mini
To solve the given system of equations graphically, we have the following two equations:
1. \( y = \frac{1}{2}x + 7 \)
2. \( y = -2x - 8 \)
### Step 1: Graph the first equation
For the equation \( y = \frac{1}{2}x + 7 \):
- The y-intercept is 7 (point (0, 7)).
- The slope is \(\frac{1}{2}\), which means for every increase of 1 in \(x\), \(y\) increases by \(\frac{1}{2}\).
You can plot additional points to help draw the line. For example:
- When \( x = 2 \):
\[
y = \frac{1}{2}(2) + 7 = 8 \quad \text{(point (2, 8))}
\]
- When \( x = -2 \):
\[
y = \frac{1}{2}(-2) + 7 = 6 \quad \text{(point (-2, 6))}
\]
### Step 2: Graph the second equation
For the equation \( y = -2x - 8 \):
- The y-intercept is -8 (point (0, -8)).
- The slope is -2, which means for every increase of 1 in \(x\), \(y\) decreases by 2.
You can plot additional points to help draw the line. For example:
- When \( x = 0 \): \( y = -8 \quad \text{(point (0, -8))} \)
- When \( x = 1 \):
\[
y = -2(1) - 8 = -10 \quad \text{(point (1, -10))}
\]
- When \( x = -1 \):
\[
y = -2(-1) - 8 = -6 \quad \text{(point (-1, -6))}
\]
### Step 3: Plot and find the intersection
Plot both lines on the same graph. The point where the two lines intersect is the solution to the system of equations.
### Intersection Coordinate
To find the intersection point algebraically:
Set the two equations equal to each other:
\[
\frac{1}{2}x + 7 = -2x - 8
\]
Multiply the entire equation by 2 to eliminate the fraction:
\[
x + 14 = -4x - 16
\]
Combine like terms:
\[
x + 4x = -16 - 14
\]
\[
5x = -30
\]
\[
x = -6
\]
Now, substitute \( x = -6 \) back into either equation to find \( y \):
Using the first equation,
\[
y = \frac{1}{2}(-6) + 7 = -3 + 7 = 4
\]
Thus, the solution to the system of equations is:
\[
\text{Intersection Point: } (-6, 4)
\]
### Conclusion
The solution to the system of equations is the point \((-6, 4)\). You can plot this on your graph to verify that it is indeed the intersection of the two lines.
1. \( y = \frac{1}{2}x + 7 \)
2. \( y = -2x - 8 \)
### Step 1: Graph the first equation
For the equation \( y = \frac{1}{2}x + 7 \):
- The y-intercept is 7 (point (0, 7)).
- The slope is \(\frac{1}{2}\), which means for every increase of 1 in \(x\), \(y\) increases by \(\frac{1}{2}\).
You can plot additional points to help draw the line. For example:
- When \( x = 2 \):
\[
y = \frac{1}{2}(2) + 7 = 8 \quad \text{(point (2, 8))}
\]
- When \( x = -2 \):
\[
y = \frac{1}{2}(-2) + 7 = 6 \quad \text{(point (-2, 6))}
\]
### Step 2: Graph the second equation
For the equation \( y = -2x - 8 \):
- The y-intercept is -8 (point (0, -8)).
- The slope is -2, which means for every increase of 1 in \(x\), \(y\) decreases by 2.
You can plot additional points to help draw the line. For example:
- When \( x = 0 \): \( y = -8 \quad \text{(point (0, -8))} \)
- When \( x = 1 \):
\[
y = -2(1) - 8 = -10 \quad \text{(point (1, -10))}
\]
- When \( x = -1 \):
\[
y = -2(-1) - 8 = -6 \quad \text{(point (-1, -6))}
\]
### Step 3: Plot and find the intersection
Plot both lines on the same graph. The point where the two lines intersect is the solution to the system of equations.
### Intersection Coordinate
To find the intersection point algebraically:
Set the two equations equal to each other:
\[
\frac{1}{2}x + 7 = -2x - 8
\]
Multiply the entire equation by 2 to eliminate the fraction:
\[
x + 14 = -4x - 16
\]
Combine like terms:
\[
x + 4x = -16 - 14
\]
\[
5x = -30
\]
\[
x = -6
\]
Now, substitute \( x = -6 \) back into either equation to find \( y \):
Using the first equation,
\[
y = \frac{1}{2}(-6) + 7 = -3 + 7 = 4
\]
Thus, the solution to the system of equations is:
\[
\text{Intersection Point: } (-6, 4)
\]
### Conclusion
The solution to the system of equations is the point \((-6, 4)\). You can plot this on your graph to verify that it is indeed the intersection of the two lines.