Question

Find the value of x if the line passing through (5,10) and (x,4) is parallel to y=6x+7

Answer 1

To determine the value of \( x \) for which the line through the points \( (5,10) \) and \( (x,4) \) is parallel to the line described by the equation \( y = 6x + 7 \), we first identify the slope of the given line \( y = 6x + 7 \).

The equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. From \( y = 6x + 7 \), we see that the slope \( m \) is \( 6 \).

Since parallel lines have the same slope, the line passing through the points \( (5, 10) \) and \( (x, 4) \) must also have a slope of \( 6 \). The slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as:

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

Substituting \( (x_1, y_1) = (5, 10) \) and \( (x_2, y_2) = (x, 4) \):

\[
m = \frac{4 - 10}{x - 5}
\]

This simplifies to:

\[
m = \frac{-6}{x - 5}
\]

We set this equal to the slope of the line \( y = 6x + 7 \):

\[
\frac{-6}{x - 5} = 6
\]

Next, we solve for \( x \). First, cross-multiply:

\[
-6 = 6(x - 5)
\]

Expanding the right side:

\[
-6 = 6x - 30
\]

Next, add \( 30 \) to both sides:

\[
24 = 6x
\]

Now, divide both sides by \( 6 \):

\[
x = 4
\]

Thus, the value of \( x \) is:

\[
\boxed{4}
\]