A given line has the following equation:
[tex]$y=3 x-5$[/tex]
Drag each equation to the correct location beneath the descriptions. Not all equations will be used.
| same as given line | parallel to given line | perpendicular to given line |
| --- | --- | --- |
| | | |
[tex]$3 x+y=-5$[/tex]
[tex]$3 x+y=15$[/tex]
[tex]$x+3 y=-15$[/tex]
[tex]$3 x-y=5$[/tex]
[tex]$3 x-y=-5$[/tex]
Answer (1)
Rewrite each equation in slope-intercept form ( y = m x + b ) to find the slope and y-intercept.
Compare the slope and y-intercept of each equation to the given line y = 3 x − 5 .
Identify equations with the same slope and y-intercept (same line), same slope but different y-intercept (parallel line), and a slope that is the negative reciprocal (perpendicular line).
Classify the equations: Same: 3 x − y = 5 , Parallel: 3 x − y = − 5 , Perpendicular: x + 3 y = − 15 .
Explanation
Understanding the Problem We are given the equation of a line y = 3 x − 5 and we need to classify the given equations as 'same as given line', 'parallel to given line', or 'perpendicular to given line'.
Rewriting Equations Let's rewrite each of the given equations in the slope-intercept form ( y = m x + b ) to easily identify their slopes and y-intercepts.
Finding Slopes and Intercepts
3 x + y = − 5 ⟹ y = − 3 x − 5 . The slope is -3 and the y-intercept is -5.
3 x + y = 15 ⟹ y = − 3 x + 15 . The slope is -3 and the y-intercept is 15.
x + 3 y = − 15 ⟹ 3 y = − x − 15 ⟹ y = − 3 1 x − 5 . The slope is − 3 1 and the y-intercept is -5.
3 x − y = 5 ⟹ − y = − 3 x + 5 ⟹ y = 3 x − 5 . The slope is 3 and the y-intercept is -5.
3 x − y = − 5 ⟹ − y = − 3 x − 5 ⟹ y = 3 x + 5 . The slope is 3 and the y-intercept is 5.
Comparing with the Given Line Now, let's compare the slope and y-intercept of each rewritten equation with the slope and y-intercept of the given line ( y = 3 x − 5 ). The given line has a slope of 3 and a y-intercept of -5.
Classifying the Equations
Same as given line: The equation must have the same slope and y-intercept. From the rewritten equations, y = 3 x − 5 (equation 4) has the same slope and y-intercept as the given line.
Parallel to given line: The equation must have the same slope but a different y-intercept. From the rewritten equations, y = 3 x + 5 (equation 5) has the same slope but a different y-intercept as the given line.
Perpendicular to given line: The equation must have a slope that is the negative reciprocal of the slope of the given line. The slope of the given line is 3, so the slope of a perpendicular line is − 3 1 . From the rewritten equations, y = − 3 1 x − 5 (equation 3) has a slope of − 3 1 .
Final Answer Therefore, the classifications are:
Same as given line: 3 x − y = 5
Parallel to given line: 3 x − y = − 5
Perpendicular to given line: x + 3 y = − 15
Examples
Understanding the relationships between lines (same, parallel, perpendicular) is crucial in various fields. For example, architects use these concepts to design buildings with parallel walls or perpendicular supports. City planners use parallel lines for roads and perpendicular lines for intersections. In computer graphics, these concepts are used to create 2D and 3D images.
