Linear Relationships and Graphing Quiz — Secondary 3

Answer: 3

The slope measures how much y changes for every one unit that x increases. The formula is a = (y₂ − y₁) / (x₂ − x₁). Take (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 11). The rise is 11 − 2 = 9 and the run is 4 − 1 = 3. So a = 9 / 3 = 3. In plain words: every time x moves one step to the right, y climbs by 3. You may take the points in the other order — (2 − 11) / (1 − 4) = −9 / −3 = 3 — and you get the same slope, which is a good way to double-check. What you must never do is mix the orders, like (11 − 2) / (1 − 4), since that flips the sign.

Answer: The value of y when x = 0

In the form y = ax + b, the letter a is the slope and b is the y-intercept — the initial value. You can see why by substituting x = 0: y = −2(0) + 5 = 5. So the line passes through the point (0, 5), which is where it crosses the vertical axis. Now the three rejected options, because each names a real quantity that simply is not 5. The slope here is −2, and the line falls two units for each step to the right. The value of x when y = 0 is the x-intercept: 0 = −2x + 5 gives x = 2.5, a different number entirely. And the change in y when x increases by 1 is just another way of saying the slope, so it is −2 as well. In a real situation, b is the starting amount — a membership fee, an initial height, a base salary — and a is what happens per unit afterwards.

Answer: 5

Substitute 3 everywhere you see x, then follow the order of operations: multiplication before subtraction. y = 4(3) − 7 = 12 − 7 = 5. So the point (3, 5) is on the line. A common error is computing 4 × (3 − 7) = −16, which would be the value for y = 4(x − 7) — a different line entirely. The rule as written multiplies x by 4 first, and only then takes 7 away.

Answer: y = 2x − 1

Read two easy points off the graph, ideally where the line crosses grid corners. The line passes through (0, −1) and (2, 3). The y-intercept is read directly: the line crosses the vertical axis at −1, so b = −1. That already rules out y = 2x + 1. Now the slope: from (0, −1) to (2, 3), y rises by 3 − (−1) = 4 while x advances by 2 − 0 = 2, so a = 4 / 2 = 2. The line climbs steeply to the right, so the slope must be positive, which rules out y = −2x − 1; and it climbs faster than one unit per step, which rules out y = x − 1. The rule is y = 2x − 1. Confirm with a third point: at x = 1, the graph shows y = 1, and 2(1) − 1 = 1. It fits.

Answer: 4

Every point on the x-axis has a height of zero, so the x-intercept is found by setting y = 0 and solving. 0 = 3x − 12. Add 12 to both sides: 12 = 3x. Divide by 3: x = 4. The line crosses the horizontal axis at the point (4, 0). Do not confuse this with the y-intercept, which you get by setting x = 0 and which here equals −12. A quick sanity check: the line starts at −12 when x = 0 and climbs 3 per step, so it needs 12 ÷ 3 = 4 steps to reach zero.

Answer: 90

Write the rule before touching the numbers. The registration fee is paid once, so it is the initial value b = 30. Each visit adds $5, so the rate of change is a = 5. The rule is C = 5n + 30, where n is the number of visits. Now substitute n = 12: C = 5(12) + 30 = 60 + 30 = 90 dollars. Two details worth noticing: the graph of this situation is a line that starts at 30 on the vertical axis rather than at the origin, and the average cost per visit, 90 ÷ 12 = 7.50, is not the same as the rate of change of 5 — the registration fee is spread over the visits.

Answer: y = (2/3)x − 4

Two lines are parallel when they have exactly the same slope. The slope here is 2/3, so look for another rule whose coefficient of x is also 2/3: that is y = (2/3)x − 4. Its y-intercept is different, so it is a genuinely distinct line running alongside the first and never meeting it. Why does the question say without being the same line? Because a line counts as parallel to itself — two identical lines are called coincident parallels — so the wording has to rule that case out, otherwise the original rule would be an acceptable answer as well. The option y = (3/2)x + 1 has the reciprocal slope, which is steeper, not parallel. The option −(2/3)x + 1 has the opposite slope, so it goes down instead of up. And y = 2x + 3 has slope 2, a completely different steepness. Worth filing away for later: perpendicular lines have slopes whose product is −1, which here would be −3/2.

Answer: 2

At the intersection point the two lines share the same x and the same y. Since both rules give y, you can set the right-hand sides equal to each other: 2x + 1 = −x + 7. Add x to both sides to gather the x terms: 3x + 1 = 7. Subtract 1: 3x = 6. Divide by 3: x = 2. To get the full point, substitute into either rule: y = 2(2) + 1 = 5, and the check with the other rule gives y = −2 + 7 = 5. Same value, so the lines meet at (2, 5). Verifying in the second equation is not optional busywork — it is what catches an arithmetic slip.

Answer: Yes, because 2(3) − 1 = 5

A point belongs to a line exactly when its coordinates make the equation true. In the pair (3, 5), the first number is always x and the second is always y, so substitute x = 3 and y = 5. The right side gives 2(3) − 1 = 6 − 1 = 5, and the left side is 5. The two match, so yes, the point is on the line. Each wrong option is a specific slip. Using 2(5) − 1 = 9 substitutes the numbers the wrong way round, putting y where x belongs — a reversal that trips up a lot of students. Using 3 + 5 = 8 adds the coordinates, which tests nothing about the rule. And 2(3) + 1 = 7 takes the right value of x but adds 1 instead of subtracting it: the arithmetic is fine, the sign is not.

Answer: −1

Find the slope first, then use one point to find b. Slope: a = (17 − 5) / (6 − 2) = 12 / 4 = 3. So the rule looks like y = 3x + b. Now substitute a known point — take (2, 5): 5 = 3(2) + b, so 5 = 6 + b, and b = −1. The rule is y = 3x − 1. Check with the other point, which is the whole reason to have two: 3(6) − 1 = 18 − 1 = 17. Correct. Reasoning backwards also works: from x = 2 to x = 0 you take two steps left, each one dropping y by 3, so y goes from 5 down to 5 − 6 = −1.

Answer: 60 km/h

On a distance-time graph, the slope is the speed, because slope is the change in distance divided by the change in time — which is exactly what speed means. Read two points on the line: it starts at the origin (0 h, 0 km) and reaches (4 h, 240 km). The rise is 240 − 0 = 240 km and the run is 4 − 0 = 4 h, so the slope is 240 / 4 = 60 km/h. Because the graph is a straight line, the speed is constant for the whole trip; a curve would mean the speed changes. And because the line goes through the origin, distance is directly proportional to time here — the rule is simply d = 60t, with no constant term.

Answer: 15 hours

Write one rule for each company, using h for the number of hours. Company A: C = 3h + 20. Company B: C = 2h + 35. "Cost the same" means the two expressions are equal: 3h + 20 = 2h + 35. Subtract 2h from both sides to collect the h terms on the left: h + 20 = 35. Subtract 20: h = 15 hours. Check both: A gives 3(15) + 20 = 45 + 20 = 65 dollars, and B gives 2(15) + 35 = 30 + 35 = 65 dollars. They match, so 15 hours is right. The wrong options each come from a mishandled step: 5 divides the 15 dollar gap by 3 instead of by the difference of the two rates, 7.5 divides it by 2, and 3 divides it by 5. Reading it as a graph makes the story clear: A starts cheaper but climbs faster, the two lines cross at (15, 65), and after that point B becomes the better deal.