What is the equation of the line that is parallel to the given line and passes through the point $(-3,2)$?

$3 x-4 y=-17$
$3 x-4 y=-20$
$4 x+3 y=-2$
$4 x+3 y=-6$

Find the slope of the given line by converting it to slope-intercept form: y = 4 3 ​ x + 4 17 ​ , so the slope is 4 3 ​ .
Use the point-slope form of a line with the point ( − 3 , 2 ) and the slope 4 3 ​ : y − 2 = 4 3 ​ ( x + 3 ) .
Convert the equation to standard form: 3 x − 4 y = − 17 .
The equation of the parallel line is 3 x − 4 y = − 17 ​ .

Explanation

Understanding the Problem We are given the equation of a line 3 x − 4 y = − 17 and a point ( − 3 , 2 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point.

Finding the Slope Parallel lines have the same slope. To find the slope of the given line, we can rewrite the equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept.

Slope-Intercept Form Let's rewrite the given equation 3 x − 4 y = − 17 in slope-intercept form:


Subtract 3 x from both sides: − 4 y = − 3 x − 17
Divide both sides by − 4 : y = 4 3 ​ x + 4 17 ​
So, the slope of the given line is 4 3 ​ .

Point-Slope Form Since the parallel line has the same slope, its slope is also 4 3 ​ . Now we can use the point-slope form of a line, which is y − y 1 ​ = m ( x − x 1 ​ ) , where ( x 1 ​ , y 1 ​ ) is the given point and m is the slope.

Applying Point-Slope Form We are given the point ( − 3 , 2 ) , so x 1 ​ = − 3 and y 1 ​ = 2 . Plugging these values and the slope m = 4 3 ​ into the point-slope form, we get:


y − 2 = 4 3 ​ ( x − ( − 3 ))
y − 2 = 4 3 ​ ( x + 3 )

Standard Form Now, let's rewrite the equation in standard form ( A x + B y = C ):

Multiply both sides by 4: 4 ( y − 2 ) = 3 ( x + 3 )
4 y − 8 = 3 x + 9
Subtract 4 y from both sides: − 8 = 3 x − 4 y + 9
Subtract 9 from both sides: − 17 = 3 x − 4 y
So, the equation of the parallel line in standard form is 3 x − 4 y = − 17 .

Final Answer The equation of the line that is parallel to the given line 3 x − 4 y = − 17 and passes through the point ( − 3 , 2 ) is 3 x − 4 y = − 17 .

Examples
Understanding parallel lines is crucial in architecture and design. For instance, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. Imagine designing a rectangular room; the opposite walls must be parallel to each other to ensure the room is functional and aesthetically pleasing. The equation of a line helps define these parallel relationships mathematically, ensuring precision in construction. In urban planning, parallel streets are often designed to optimize traffic flow and create organized city layouts. The concept of parallel lines and their equations is fundamental in creating orderly and efficient environments.

Answered by GinnyAnswer | 2025-07-05