Writing the equation of a straight line
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All 16 Flashcards — Writing the equation of a straight line
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Question
What is the slope-intercept form of a straight line?
Answer
y = mx + c m = gradient (slope), c = y-intercept. This form directly shows both key features of the line.
Question
Write the equation of a line with gradient 5 and y-intercept −3.
Answer
Substitute directly into y = mx + c: y = 5x − 3. The gradient goes with x; the y-intercept is the constant.
Question
A line has equation y = −(1/2)x + 6. Write down the gradient and y-intercept and describe the direction of the line.
Answer
Gradient m = −1/2. y-intercept c = 6. The line starts high on the y-axis and falls gently — it goes down 1 for every 2 units to the right.
Question
Exam trap: A student writes the equation of a line as "m = 3, c = 7" and stops. What must they write instead?
Answer
IB always requires a full equation, not just the values of m and c. Write: y = 3x + 7. The equation must start with "y =" and show both m and c in the correct form.
Question
Describe the method for finding the equation of a line given its gradient and one point on the line.
Answer
1. Write y = mx + c with the known gradient m. 2. Substitute the coordinates of the given point for x and y. 3. Solve for c. 4. Write the full equation with both m and c.
Question
Find the equation of the line with gradient 3 that passes through (2, 8).
Answer
y = 3x + c. Substitute (2, 8): 8 = 3(2) + c → 8 = 6 + c → c = 2. Equation: y = 3x + 2.
Question
Find the equation of the line with gradient −2 that passes through (−1, 5).
Answer
y = −2x + c. Substitute (−1, 5): 5 = −2(−1) + c → 5 = 2 + c → c = 3. Equation: y = −2x + 3. Check: plug in x = −1: y = −2(−1) + 3 = 5 ✓
Question
Exam trap: A student finds c = 4 but writes the final equation as y = mx + 4 without substituting m. What is the issue?
Answer
They left m as a letter instead of replacing it with the actual gradient value. If gradient = 2 and c = 4, the equation must be: y = 2x + 4. Always replace m with its value in the final answer.
Question
What are the two steps to find the equation of a line through two given points?
Answer
Step 1: Calculate the gradient using m = (y₂ − y₁)/(x₂ − x₁). Step 2: Use one point and the gradient to find c (substitute into y = mx + c).
Question
Find the equation of the line through (1, 4) and (3, 10).
Answer
m = (10 − 4)/(3 − 1) = 6/2 = 3. y = 3x + c. Use (1, 4): 4 = 3(1) + c → c = 1. Equation: y = 3x + 1.
Question
Find the equation of the line through (0, −3) and (4, 5).
Answer
m = (5 − (−3))/(4 − 0) = 8/4 = 2. y-intercept: when x = 0, y = −3, so c = −3 directly. Equation: y = 2x − 3. Shortcut: if one point is the y-intercept (x = 0), c = that y-value immediately.
Question
Exam trap: A student uses two points to find the gradient m = 4, then writes y = 4x without finding c. What must they still do?
Answer
They must use one of the given points to substitute into y = 4x + c and solve for c. The equation y = 4x only works if the line passes through the origin — that must be verified, not assumed.
Question
What is the general form of a straight line equation?
Answer
ax + by + d = 0 (sometimes written ax + by = c). All terms are moved to one side, leaving zero on the other. IB accepts both y = mx + c and general form unless the question specifies which.
Question
Rearrange y = 3x − 5 into the form ax + by + d = 0 with integer coefficients.
Answer
Move all terms to the left: 3x − y − 5 = 0. Or equivalently: −3x + y + 5 = 0 (both are valid; IB usually wants positive leading coefficient).
Question
Convert 2x − y + 8 = 0 back into y = mx + c form and state the gradient and y-intercept.
Answer
Rearrange: y = 2x + 8. Gradient m = 2, y-intercept c = 8.
Question
Exam trap: A question asks for the equation of a line "in the form ax + by + d = 0." A student writes y = 2x − 4. How many marks will they lose?
Answer
IB requires the specific form asked for. Leaving it as y = 2x − 4 does not match ax + by + d = 0. Correct: 2x − y − 4 = 0. Always re-read what form the question requires before writing the final answer.
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Full study notes for Writing the equation of a straight line
Topic 2.1 hub
Equations of a line
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