Question
Graph and then find the solution for
and
.
Point
Line
Edit Label
Undo
Redo
Reset
Delete
0 12345678910-1-2-3-4-5-6-7-8-9-1012345678910-1-2-3-4-5-6-7-8-9-10
(b)
The solution to
y
=
1
2
x
+
2
y=
2
1
x+2 and
y
=
−
1
4
x
+
8
y=−
4
1
x+8 is the point
(
,
)
and
.
Point
Line
Edit Label
Undo
Redo
Reset
Delete
0 12345678910-1-2-3-4-5-6-7-8-9-1012345678910-1-2-3-4-5-6-7-8-9-10
(b)
The solution to
y
=
1
2
x
+
2
y=
2
1
x+2 and
y
=
−
1
4
x
+
8
y=−
4
1
x+8 is the point
(
,
)
Answers
GPT-4o mini
To find the solution to the given system of equations, we can start by rewriting them for clarity.
1. \( y = \frac{1}{2} x + 2 \)
2. \( y = -\frac{1}{4} x + 8 \)
**Step 1: Graph the equations**
1. **Graphing \( y = \frac{1}{2} x + 2 \)**:
- The y-intercept is 2 (when \( x = 0 \), \( y = 2 \)).
- The slope is \( \frac{1}{2} \), meaning for every increase of 1 in \( x \), \( y \) increases by \( \frac{1}{2} \).
- Plot points:
- For \( x = 0 \): \( (0, 2) \)
- For \( x = 2 \): \( y = \frac{1}{2}(2) + 2 = 3 \) (point \( (2, 3) \))
- For \( x = 4 \): \( y = \frac{1}{2}(4) + 2 = 4 \) (point \( (4, 4) \))
2. **Graphing \( y = -\frac{1}{4} x + 8 \)**:
- The y-intercept is 8 (when \( x = 0 \), \( y = 8 \)).
- The slope is \(-\frac{1}{4}\), meaning for every increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{4} \).
- Plot points:
- For \( x = 0 \): \( (0, 8) \)
- For \( x = 4 \): \( y = -\frac{1}{4}(4) + 8 = 7 \) (point \( (4, 7) \))
- For \( x = 8 \): \( y = -\frac{1}{4}(8) + 8 = 6 \) (point \( (8, 6) \))
**Step 2: Find the intersection**
Set the right-hand sides of the equations equal to each other to find the intersection point:
\[
\frac{1}{2} x + 2 = -\frac{1}{4} x + 8
\]
**Step 3: Solve for \( x \)**
1. Multiply the entire equation by 4 to eliminate the fractions:
\[
4\left(\frac{1}{2} x + 2\right) = 4\left(-\frac{1}{4} x + 8\right)
\]
This simplifies to:
\[
2x + 8 = -x + 32
\]
2. Rearranging gives:
\[
2x + x = 32 - 8
\]
\[
3x = 24
\]
\[
x = 8
\]
**Step 4: Substitute \( x \) back to find \( y \)**
Using the first equation:
\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]
The point of intersection, which is the solution to the system of equations, is:
\[
(8, 6)
\]
**Conclusion:**
The solution to the system of equations is the point \( (8, 6) \).
1. \( y = \frac{1}{2} x + 2 \)
2. \( y = -\frac{1}{4} x + 8 \)
**Step 1: Graph the equations**
1. **Graphing \( y = \frac{1}{2} x + 2 \)**:
- The y-intercept is 2 (when \( x = 0 \), \( y = 2 \)).
- The slope is \( \frac{1}{2} \), meaning for every increase of 1 in \( x \), \( y \) increases by \( \frac{1}{2} \).
- Plot points:
- For \( x = 0 \): \( (0, 2) \)
- For \( x = 2 \): \( y = \frac{1}{2}(2) + 2 = 3 \) (point \( (2, 3) \))
- For \( x = 4 \): \( y = \frac{1}{2}(4) + 2 = 4 \) (point \( (4, 4) \))
2. **Graphing \( y = -\frac{1}{4} x + 8 \)**:
- The y-intercept is 8 (when \( x = 0 \), \( y = 8 \)).
- The slope is \(-\frac{1}{4}\), meaning for every increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{4} \).
- Plot points:
- For \( x = 0 \): \( (0, 8) \)
- For \( x = 4 \): \( y = -\frac{1}{4}(4) + 8 = 7 \) (point \( (4, 7) \))
- For \( x = 8 \): \( y = -\frac{1}{4}(8) + 8 = 6 \) (point \( (8, 6) \))
**Step 2: Find the intersection**
Set the right-hand sides of the equations equal to each other to find the intersection point:
\[
\frac{1}{2} x + 2 = -\frac{1}{4} x + 8
\]
**Step 3: Solve for \( x \)**
1. Multiply the entire equation by 4 to eliminate the fractions:
\[
4\left(\frac{1}{2} x + 2\right) = 4\left(-\frac{1}{4} x + 8\right)
\]
This simplifies to:
\[
2x + 8 = -x + 32
\]
2. Rearranging gives:
\[
2x + x = 32 - 8
\]
\[
3x = 24
\]
\[
x = 8
\]
**Step 4: Substitute \( x \) back to find \( y \)**
Using the first equation:
\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]
The point of intersection, which is the solution to the system of equations, is:
\[
(8, 6)
\]
**Conclusion:**
The solution to the system of equations is the point \( (8, 6) \).
6 is wrong
GPT-4o mini
I apologize for any confusion in my previous calculations. Let’s carefully go through the system of equations again step-by-step.
The equations to solve are:
1. \( y = \frac{1}{2}x + 2 \)
2. \( y = -\frac{1}{4}x + 8 \)
**Step 1: Set the equations equal to each other**
Since both equations are equal to \( y \), we can set them equal to find \( x \):
\[
\frac{1}{2}x + 2 = -\frac{1}{4}x + 8
\]
**Step 2: Solve for \( x \)**
To eliminate the fractions, we can multiply the entire equation by 4:
\[
4\left(\frac{1}{2}x + 2\right) = 4\left(-\frac{1}{4}x + 8\right)
\]
This simplifies to:
\[
2x + 8 = -x + 32
\]
Next, we rearrange the equation to isolate \( x \):
\[
2x + x = 32 - 8
\]
\[
3x = 24
\]
\[
x = 8
\]
**Step 3: Substitute \( x \) back into one of the equations to find \( y \)**
Let's substitute \( x = 8 \) back into the first equation:
\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]
Now let's verify by substituting \( x = 8 \) into the second equation:
\[
y = -\frac{1}{4}(8) + 8 = -2 + 8 = 6
\]
Both equations give us \( y = 6 \).
**Conclusion:**
The point of intersection for the two lines, which is the solution to the system of equations, is indeed:
\[
(8, 6)
\]
If you believe there is an error or if you would like to discuss further, please let me know!
The equations to solve are:
1. \( y = \frac{1}{2}x + 2 \)
2. \( y = -\frac{1}{4}x + 8 \)
**Step 1: Set the equations equal to each other**
Since both equations are equal to \( y \), we can set them equal to find \( x \):
\[
\frac{1}{2}x + 2 = -\frac{1}{4}x + 8
\]
**Step 2: Solve for \( x \)**
To eliminate the fractions, we can multiply the entire equation by 4:
\[
4\left(\frac{1}{2}x + 2\right) = 4\left(-\frac{1}{4}x + 8\right)
\]
This simplifies to:
\[
2x + 8 = -x + 32
\]
Next, we rearrange the equation to isolate \( x \):
\[
2x + x = 32 - 8
\]
\[
3x = 24
\]
\[
x = 8
\]
**Step 3: Substitute \( x \) back into one of the equations to find \( y \)**
Let's substitute \( x = 8 \) back into the first equation:
\[
y = \frac{1}{2}(8) + 2 = 4 + 2 = 6
\]
Now let's verify by substituting \( x = 8 \) into the second equation:
\[
y = -\frac{1}{4}(8) + 8 = -2 + 8 = 6
\]
Both equations give us \( y = 6 \).
**Conclusion:**
The point of intersection for the two lines, which is the solution to the system of equations, is indeed:
\[
(8, 6)
\]
If you believe there is an error or if you would like to discuss further, please let me know!