Question

Suppose the line through points (x,6) and (1,2) is parallel to the graph of 2x + y = 3. Find the value of x. Show your work.

Answer 1

To solve the problem, we need to determine the value of \( x \) such that the line through the points \( (x, 6) \) and \( (1, 2) \) is parallel to the line represented by the equation \( 2x + y = 3 \).

### Step 1: Find the slope of the given line \( 2x + y = 3 \).

To find the slope, we first rewrite the equation in slope-intercept form \( y = mx + b \):

\[
y = -2x + 3
\]

The slope (m) of this line is \( -2 \).

### Step 2: Determine the slope of the line through the points \( (x, 6) \) and \( (1, 2) \).

To find the slope of the line through the points \( (x, 6) \) and \( (1, 2) \), we use the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \( (x_1, y_1) = (x, 6) \) and \( (x_2, y_2) = (1, 2) \). Substituting in the coordinates, we have:

\[
m = \frac{2 - 6}{1 - x} = \frac{-4}{1 - x}
\]

### Step 3: Set the slopes equal to each other.

Since the two lines are parallel, their slopes must be equal:

\[
\frac{-4}{1 - x} = -2
\]

### Step 4: Solve for \( x \).

To eliminate the negative signs, we can multiply both sides by -1:

\[
\frac{4}{1 - x} = 2
\]

Next, we cross-multiply:

\[
4 = 2(1 - x)
\]

Expanding the right side gives us:

\[
4 = 2 - 2x
\]

Now we solve for \( x \):

\[
4 - 2 = -2x
\]
\[
2 = -2x
\]
\[
x = -1
\]

### Conclusion

Thus, the value of \( x \) is

\[
\boxed{-1}
\]