Question
Solve each system by graphing. Tell whether the system has one solution, infinitely many, or no solutions.
Y = 3x + 2
Y= -1/2x + 9
Y = 3x + 2
Y= -1/2x + 9
Answers
GPT-4o mini
To solve the system of equations by graphing, let's graph each equation step by step.
1. **Graphing the first equation:**
\[
y = 3x + 2
\]
- The y-intercept is 2. This is the point (0, 2).
- The slope is 3, which means for every 1 unit you move to the right (positive x-direction), you move 3 units up (positive y-direction).
- From (0, 2), if we move 1 unit right to (1, 2 + 3) = (1, 5), we can plot that point.
- We can also move in the opposite direction: from (0, 2) go left 1 unit (to -1, 2) and down 3 units (to -1, -1), which gives us the point (-1, -1).
The line can then be drawn through points (0, 2) and (1, 5), and extended in both directions.
2. **Graphing the second equation:**
\[
y = -\frac{1}{2}x + 9
\]
- The y-intercept is 9. This is the point (0, 9).
- The slope is -1/2, which means for every 1 unit to the right (positive x-direction), you move 1/2 unit down (negative y-direction).
- From (0, 9), if we move 1 unit right (to 1), we move down 0.5 units to (1, 8.5).
- We can also move in the opposite direction: from (0, 9) go left 1 unit (to -1, 9) and down 0.5 units (to -1, 8.5), which gives us the point (-1, 8.5).
The line can then be drawn through points (0, 9) and (1, 8.5), and extended in both directions.
3. **Finding the intersection:**
After both lines are graphed, we check where they intersect. The first line is increasing with a steep slope, while the second line decreases gently.
By observing the trend of both lines (one rising steeply and the other falling gently), you can expect them to intersect in one place.
### Conclusion
- The two lines intersect at a single point. Hence, the system has **one solution**.
To find the exact coordinates of the intersection point, we can solve the equations algebraically:
Set the two equations equal to each other:
\[
3x + 2 = -\frac{1}{2}x + 9
\]
Combine like terms to solve for \( x \):
1. Add \(\frac{1}{2}x\) to both sides:
\[
3x + \frac{1}{2}x + 2 = 9
\]
\[
\frac{6}{2}x + \frac{1}{2}x + 2 = 9
\]
\[
\frac{7}{2}x + 2 = 9
\]
2. Subtract 2 from both sides:
\[
\frac{7}{2}x = 7
\]
3. Multiply by \(\frac{2}{7}\):
\[
x = 2
\]
Plug \( x = 2 \) back into one of the original equations to find \( y \):
Using the first equation:
\[
y = 3(2) + 2 = 6 + 2 = 8.
\]
Thus, the intersection point is \( (2, 8) \). The system has **one solution: \((2, 8)\)**.
1. **Graphing the first equation:**
\[
y = 3x + 2
\]
- The y-intercept is 2. This is the point (0, 2).
- The slope is 3, which means for every 1 unit you move to the right (positive x-direction), you move 3 units up (positive y-direction).
- From (0, 2), if we move 1 unit right to (1, 2 + 3) = (1, 5), we can plot that point.
- We can also move in the opposite direction: from (0, 2) go left 1 unit (to -1, 2) and down 3 units (to -1, -1), which gives us the point (-1, -1).
The line can then be drawn through points (0, 2) and (1, 5), and extended in both directions.
2. **Graphing the second equation:**
\[
y = -\frac{1}{2}x + 9
\]
- The y-intercept is 9. This is the point (0, 9).
- The slope is -1/2, which means for every 1 unit to the right (positive x-direction), you move 1/2 unit down (negative y-direction).
- From (0, 9), if we move 1 unit right (to 1), we move down 0.5 units to (1, 8.5).
- We can also move in the opposite direction: from (0, 9) go left 1 unit (to -1, 9) and down 0.5 units (to -1, 8.5), which gives us the point (-1, 8.5).
The line can then be drawn through points (0, 9) and (1, 8.5), and extended in both directions.
3. **Finding the intersection:**
After both lines are graphed, we check where they intersect. The first line is increasing with a steep slope, while the second line decreases gently.
By observing the trend of both lines (one rising steeply and the other falling gently), you can expect them to intersect in one place.
### Conclusion
- The two lines intersect at a single point. Hence, the system has **one solution**.
To find the exact coordinates of the intersection point, we can solve the equations algebraically:
Set the two equations equal to each other:
\[
3x + 2 = -\frac{1}{2}x + 9
\]
Combine like terms to solve for \( x \):
1. Add \(\frac{1}{2}x\) to both sides:
\[
3x + \frac{1}{2}x + 2 = 9
\]
\[
\frac{6}{2}x + \frac{1}{2}x + 2 = 9
\]
\[
\frac{7}{2}x + 2 = 9
\]
2. Subtract 2 from both sides:
\[
\frac{7}{2}x = 7
\]
3. Multiply by \(\frac{2}{7}\):
\[
x = 2
\]
Plug \( x = 2 \) back into one of the original equations to find \( y \):
Using the first equation:
\[
y = 3(2) + 2 = 6 + 2 = 8.
\]
Thus, the intersection point is \( (2, 8) \). The system has **one solution: \((2, 8)\)**.