Physics and Mathematics
Example – Find the Slope and Express Equation in Intercept Form
Practice Questions with Step-by-Step Solutions
Question 1. Find the slope of the line given by [y = 3x − 7].
Step-by-Step Solution:
Compare with slope–intercept form [y = mx + c]
Coefficient of [x] is [3]
Conclusion:
Slope of the line is [m = 3].
Question 2. Find the slope of the line [y = −5x + 4].
Step-by-Step Solution:
Given equation is in the form [y = mx + c]
Coefficient of [x] is [−5]
Conclusion:
Slope is [m = −5].
Question 3. Find the slope of the line [y = 6].
Step-by-Step Solution:
Equation has no [x] term
It represents a line parallel to the x-axis
Conclusion:
Slope is [m = 0].
Question 4. Find the slope of the line given by [2x + 3y − 6 = 0].
Step-by-Step Solution:
Compare with general form [Ax + By + C = 0]
Here [A = 2], [B = 3]
Slope formula:
[m = −\dfrac{A}{B}]
Substitute values:
[m = −\dfrac{2}{3}]
Conclusion:
Slope is [−\dfrac{2}{3}].
Question 5. Find the slope of the straight line [5y − 10x + 15 = 0].
Step-by-Step Solution:
Rearrange into standard order:
[−10x + 5y + 15 = 0]
Identify [A = −10], [B = 5]
Slope formula:
[m = −\dfrac{A}{B}]
Substitute:
[m = −\dfrac{−10}{5} = 2]
Conclusion:
Slope of the line is [m = 2].
Question 6. Find the slope of the line given by [\dfrac{x}{3} + \dfrac{y}{6} = 1].
Step-by-Step Solution:
Given equation is in intercept form
Compare with [\dfrac{x}{a} + \dfrac{y}{b} = 1]
Here [a = 3], [b = 6]
Slope formula:
[m = −\dfrac{b}{a}]
Substitute:
[m = −\dfrac{6}{3} = −2]
Conclusion:
Slope of the line is [m = −2].
Question 7. Find the slope of the line [x = −4].
Step-by-Step Solution:
Equation is of the form [x = constant]
Such a line is parallel to the y-axis
Change in [x] is zero, so slope is undefined
Conclusion:
Slope of the line is not defined.
Question 8. Find the slope of the line [3y = 2x − 9].
Step-by-Step Solution:
Convert into slope–intercept form:
[y = \dfrac{2}{3}x − 3]
Compare with [y = mx + c]
Coefficient of [x] is slope
Conclusion:
Slope is [m = \dfrac{2}{3}].
Question 9. Find the slope of the line [7 − 4y = 2x].
Step-by-Step Solution:
Rearrange equation:
[−4y = 2x − 7]
Divide both sides by [−4]:
[y = −\dfrac{1}{2}x + \dfrac{7}{4}]
Identify slope from [y = mx + c]
Conclusion:
Slope is [m = −\dfrac{1}{2}].
Question 10. Without rearranging, find the slope of the line [4x − 5y + 1 = 0].
Step-by-Step Solution:
Compare with general form [Ax + By + C = 0]
Identify [A = 4], [B = −5]
Slope formula:
[m = −\dfrac{A}{B}]
Substitute values:
[m = −\dfrac{4}{−5} = \dfrac{4}{5}]
Conclusion:
Slope of the line is [m = \dfrac{4}{5}].
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