Slope & the Forms of a Line — Challenge
Three ways to write the same line. Each one is best at answering a different question.
What these problems cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Integer — A whole number, positive, negative, or zero.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Slope — How much y changes for each 1 that x increases.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Slope-intercept form — y = mx + b, with m the slope and b the y-intercept.
- x-intercept — The x value when y = 0 — where it crosses the x-axis.
- Point-slope form — y - y1 = m(x - x1). Best when you know a point and the slope.
- Standard form — Ax + By = C. Best for finding intercepts fast.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
Problem 1
Which equation is written in slope-intercept form?
- y=mx+b
- Ax+By=C
- y-y1=m(x-x1)
- y=ax2
- x=my+b
Worked answer
y=mx+b
Slope-intercept form gives y on its own, with the slope multiplying x and the intercept added.
That is y = mx+b.
Ax+By=C is standard form and y-y1 = m(x-x1) is point-slope form; both describe lines but are arranged differently.
Problem 2
Write y=2x+3 in standard form Ax+By=C.
- 2x+y=3
- y-2x=-3
- x-2y=3
- 2x-y=-3
- 2x-y=3
Worked answer
2x-y=-3
Standard form needs both variable terms on the left and the constant on the right.
Subtract y from both sides and subtract 3: 2x-y = -3.
Check with x=0: the equation gives -y = -3, so y = 3, matching the original intercept.
Problem 3
Write the equation of the line through (2,5) with slope 3 in point-slope form.
- y-2=3(x-5)
- y-5=3(x-2)
- y+5=3(x+2)
- y=3x-5
- y-3=5(x-2)
Worked answer
y-5=3(x-2)
Point-slope form is y-y1 = m(x-x1), with the point's coordinates subtracted.
Here x1 = 2, y1 = 5 and m = 3, giving y-5 = 3(x-2).
Swapping the coordinates gives y-2 = 3(x-5), which is a different line.
Problem 4
Write y-4=2(x-1) in slope-intercept form.
- y=2x+4
- y=2x-2
- y=2x+2
- y=2x+6
- y=2x
Worked answer
y=2x+2
Expand the right side: y-4 = 2x-2.
Add 4 to both sides: y = 2x+2.
Check with x=1: y = 4, which matches the point in the original equation.
Problem 5
What is the y-intercept of y=-3x+7?
- -3
- 3
- 7
- -7
- 0
Worked answer
7
In slope-intercept form the constant added at the end is the y-intercept.
So the intercept is 7, and the slope is -3.
Substituting x=0 confirms it: y = 7.
Problem 6
Write 3x+y=9 in slope-intercept form.
- y=3x+9
- y=-3x+9
- y=-3x-9
- y=3x-9
- y=9x-3
Worked answer
y=-3x+9
Subtract 3x from both sides to leave y alone.
y = -3x+9.
The slope is -3, not 3: moving the term across the equals sign changes its sign.
Problem 7
What is the slope of y=x2-1?
- 12
- 2
- -1
- 1
- -12
Worked answer
12
Writing x2 as 12x makes the form y = mx+b visible.
The slope is therefore 12, and the y-intercept is -1.
Reading the 2 from the denominator as the slope is the trap here.
Problem 8
What is the equation of the line through (0,-2) with slope 4?
- y=4x+2
- y=-2x+4
- y=4x-2
- y=4x
- y=-4x-2
Worked answer
y=4x-2
The point (0,-2) lies on the y-axis, so the y-intercept is -2.
With slope 4 the equation is y = 4x-2.
Check with x=0: y = -2, as required.