What is the equation of the line that is parallel to the line $5x + 2y = 12$ and passes through the point $(-2, 4)$?
A. $y=-\frac{5}{2} x-1$
B. $y=-\frac{5}{2} x+5$
C. $y=\frac{2}{5} x-1$
D. $y=\frac{2}{5} x+5
Answer (2)
Find the slope of the given line 5 x + 2 y = 12 , which is − 2 5 .
Use the point-slope form of a line y − y 1 = m ( x − x 1 ) with the point ( − 2 , 4 ) and slope − 2 5 .
Substitute the values to get y − 4 = − 2 5 ( x + 2 ) .
Rewrite the equation in slope-intercept form to get the final answer: y = − 2 5 x − 1 .
Explanation
Understanding the Problem We are given a line 5 x + 2 y = 12 and a point ( − 2 , 4 ) . 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 of the Given Line First, let's find the slope of the given line. We can rewrite the equation 5 x + 2 y = 12 in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Subtracting 5 x from both sides gives 2 y = − 5 x + 12 . Dividing both sides by 2, we get y = − 2 5 x + 6 . So, the slope of the given line is − 2 5 .
Using the Point-Slope Form Since parallel lines have the same slope, the slope of the line we are looking for is also − 2 5 . Now we can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point. In our case, m = − 2 5 and ( x 1 , y 1 ) = ( − 2 , 4 ) . Plugging these values into the point-slope form, we get y − 4 = − 2 5 ( x − ( − 2 )) , which simplifies to y − 4 = − 2 5 ( x + 2 ) .
Rewriting in Slope-Intercept Form Now, let's rewrite the equation in slope-intercept form. We have y − 4 = − 2 5 ( x + 2 ) . Distributing the − 2 5 on the right side, we get y − 4 = − 2 5 x − 5 . Adding 4 to both sides, we get y = − 2 5 x − 5 + 4 , which simplifies to y = − 2 5 x − 1 .
Final Answer Therefore, the equation of the line that is parallel to the line 5 x + 2 y = 12 and passes through the point ( − 2 , 4 ) is y = − 2 5 x − 1 .
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. If a designer needs to create a roof that is parallel to the ground, they would use the concept of parallel lines to determine the slope and ensure the roof is evenly aligned. This principle helps in creating aesthetically pleasing and structurally sound buildings. In urban planning, parallel streets are often designed to facilitate efficient traffic flow and create organized city layouts. The equation of parallel lines helps in mapping and designing these layouts accurately.
The equation of the line that is parallel to 5 x + 2 y = 12 and passes through the point ( − 2 , 4 ) is y = − 2 5 x − 1 , which corresponds to option A.
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