How do you find an equation parallel to $x = 7$ and passing through $\left( {4, - 7} \right)$?

Hint: The above question is a simple question of linear equations in two variables. The general equation of the slope-intercept form of the line is given as \[y = mx + c\], where m is the slope of the line and c is the y-intercept of the line. Also, note that when a line is parallel to the x-axis then its slope is equal to 0, so the equation of such line is given as \[y = a\] where a is the y-intercept that line.

Complete step-by-step solution:
We can see from the question that we are provided with a line that is parallel to the x-axis.
Since we know that when a line is parallel to the x-axis then its slope is equal to 0.
Also, we know that the slope-intercept form of the line is given by \[y = mx + c\], where m is the slope of the line and c is the y-intercept of the line.
So, we can say that equation of the line is equal to,
$ \Rightarrow y = 0 \times x + c$
Simplify the terms,
$ \Rightarrow y = c$
Now, we know from the question that the line \[y = c\] passes through the point (4, -7).
So, the point (4, -7) will satisfy the line \[y = c\]. Substitute the value in $y = c$.
$ \Rightarrow - 7 = c$

Hence, \[y = - 7\] is the equation of the line passing through the point (4, -7) which is parallel to the x-axis.

Note: Students are required to note that the general equation of the line is given as $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where m is the slope of the line and $\left( {{y_1},{x_1}} \right)$ is the point through which the line passes. We know that the line given in the question is parallel to the x-axis, so the slope of the line is equal to 0 and since the line passes through the point (1, 7), hence the equation of line will be:
$ \Rightarrow y - \left( { - 7} \right) = 0\left( {x - 4} \right)$
Simplify the terms,
$ \Rightarrow y = - 7$
Hence, \[y = - 7\] is the equation of the line.