Question

Graph and then find the solution for
and
.

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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



(
,
)

Answer 1

6 is wrong

Answer 2

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 \( (8, 6) \).

Answer 3

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!