Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=6 x+9,(5,-3)$$

Short answer

The equation of the line that is parallel to \(y = 6x + 9\) and passes through the point (5,-3) is \(y = 6x - 33\).

Step-by-step solution

Identifying and Setting up the Equation

Start by writing the equation of the line in y = mx + b form. Since we know that parallel lines have the same slope, the slope m of the new line will be the same as the slope of the given line, which is 6. Write the equation of the new line as \(y = 6x + b\), where b is the y-intercept we need to find.

Finding the y-intercept

To find b, substitute the point (5, -3) into the equation. This point lies on the line, so it must satisfy the line equation. Therefore, the coordinates x = 5 and y = -3 can replace x and y in the equation, so it becomes \(-3 = 6(5) + b\).

Solve for b

Solve the equation \(-3 = 30 + b\), rearrange for b to obtain \(b = -3 - 30\). Therefore, b = -33.

Write Final Equation

The final step is to write down the equation of the line with the values of m (slope) and b (y-intercept) determined. The equation of the line is therefore \(y = 6x - 33\).