Questions: Writing Linear Equations

Question 1

A line has slope 3 and passes through the point (4, 7). The y-intercept is unknown. What is the most efficient first step to write the equation?

• A. Write y = 3x + b and solve for b by substituting x = 4 and y = 7
• B. Write y − 7 = 3(x − 4) using point-slope form directly
• C. Plot the point and count slope triangles back to the y-axis
• D. Use the two-point formula after creating a second point by moving one unit right

Answer: (B)

When you have a slope and a non-y-intercept point, point-slope form y − y₁ = m(x − x₁) is the direct, efficient choice — plug in the known values and you have the equation immediately. Option A also works, but it requires an extra step: substituting to find b, then rewriting. Option B is slower (you have to invent a second point). The common mistake is defaulting to slope-intercept form (y = mx + b) and treating the y-intercept as unknown — that approach works but is less direct when the y-intercept isn't given information.

Question 2

A taxi charges $2.50 per mile plus a $3.00 base fare. A student writes the equation cost = 2.50m + 3.00. What does the 3.00 represent in the context of linear equations?

• A. The slope, because it is the starting value of the fare
• B. The y-intercept, because it is the cost when miles driven equals zero
• C. A constant that adjusts the units from miles to dollars
• D. The x-intercept, because it is paid before any miles are driven

Answer: (B)

In the equation y = mx + b, b is the y-intercept — the value of y when x = 0. Here, when miles = 0, cost = $3.00 (the base fare). So 3.00 is the y-intercept, representing the initial value before any distance is driven. The slope 2.50 is the rate of change (cost per mile). A common word-problem error is identifying the initial flat fee as the 'slope' because it is 'the starting point' — but slope is always a rate of change (how much y changes per unit of x), not an initial value.

Question 3

Given only two points on a line, it is impossible to write the equation without first calculating the slope.

• T. True
• F. False

Answer: True

This is true. A line is determined by two pieces of information, but 'two points' does not directly give you slope or intercept — you must extract slope first using m = (y₂ − y₁)/(x₂ − x₁). Only after computing slope can you use either point in point-slope form to write the full equation. There is no shortcut that skips the slope calculation when given two arbitrary points. This is why the process is: compute slope → then write the equation using slope and one point.

Question 4

To write the equation of a line, you is expected to usually determine the y-intercept first.

• T. True
• F. False

Answer: False

Point-slope form y − y₁ = m(x − x₁) lets you write a valid linear equation using any known point on the line — the y-intercept is not required. If you have slope 4 and the point (3, 11), you can immediately write y − 11 = 4(x − 3) without ever computing the y-intercept. You can then simplify to slope-intercept form afterward if needed, but the y-intercept is an intermediate result, not a prerequisite. Forcing yourself to find b first when it isn't given is a common inefficiency that often leads to extra arithmetic errors.

Question 5

A problem gives you only the slope and one point that is NOT on the y-axis. Explain which equation form you should start with and why.

Model answer: Use point-slope form: y − y₁ = m(x − x₁). Plug in the known slope for m and the known point coordinates for x₁ and y₁. This form is designed exactly for this situation — you have a slope and a point, so you can write the equation directly without any additional steps.

Slope-intercept form (y = mx + b) requires knowing b, the y-intercept. If the given point is not the y-intercept, you don't know b yet. You could solve for it by substituting into y = mx + b, but that's an extra step. Point-slope form skips that step entirely: it accepts any point, not just the y-intercept, as input. This is why mathematicians invented it — it matches the most common situation in which a line is specified (slope + a point you happen to know).