Select the correct answer from each drop-down menu. Consider the given equation: [tex]3 x+2 y=8[/tex]. The equation [tex]y=3 / 2 v x+[ /tex] represents the line parallel to the given equation and passes through the point (-2,5).
Answer (2)
The slope of the given equation 3 x + 2 y = 8 is − 2 3 , which means the parallel line will also have this slope. After substituting the point ( − 2 , 5 ) into the equation of the line, we find that v = − 1 and c = 2 . Thus, the final equation of the line is y = − 2 3 x + 2 .
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Rewrite the given equation in slope-intercept form to find the slope: y = − 2 3 x + 4 . The slope is − 2 3 .
Since the parallel line has the same slope, set 2 3 v = − 2 3 and solve for v , which gives v = − 1 .
Substitute the point ( − 2 , 5 ) into the equation y = − 2 3 x + c to find c : 5 = − 2 3 ( − 2 ) + c , so c = 2 .
The equation of the line is y = − 2 3 x + 2 , so the values are v = − 1 and c = 2 . Therefore, the answer is v = − 1 , c = 2 .
Explanation
Understanding the Problem We are given the equation 3 x + 2 y = 8 and asked to find the equation of a line parallel to it that passes through the point ( − 2 , 5 ) . The equation should be in the form y = 2 3 vx + c , where we need to find the values of v and c .
Finding the Slope First, let's rewrite the given equation in slope-intercept form ( y = m x + b ) to find its slope. We have 3 x + 2 y = 8 . Subtracting 3 x from both sides gives 2 y = − 3 x + 8 . Dividing by 2, we get y = − 2 3 x + 4 . The slope of this line is − 2 3 .
Determining the Value of v Since parallel lines have the same slope, the slope of the parallel line we are looking for is also − 2 3 . We are given that the equation of the parallel line is in the form y = 2 3 vx + c . Therefore, we must have 2 3 v = − 2 3 . Solving for v , we get v = − 1 .
Finding the Value of c Now we have the equation y = 2 3 ( − 1 ) x + c , which simplifies to y = − 2 3 x + c . We know that the line passes through the point ( − 2 , 5 ) . Substituting x = − 2 and y = 5 into the equation, we get 5 = − 2 3 ( − 2 ) + c . This simplifies to 5 = 3 + c . Solving for c , we get c = 5 − 3 = 2 .
Final Equation Therefore, the equation of the line is y = − 2 3 x + 2 . Since the equation is in the form y = 2 3 vx + c , we have v = − 1 and c = 2 .
Examples
Understanding parallel lines is crucial in various fields, such as architecture and urban planning. For instance, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. Similarly, in urban planning, parallel streets can optimize traffic flow and create organized city layouts. The principles of parallel lines also extend to computer graphics, where they are used to create realistic perspectives and 3D models.
