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AQA GCSE · Question 17 · Algebra
Straight line LM has equation y = 4x - 7
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Straight line ST has equation y = (9 - x) / 4
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Are the lines LM and ST perpendicular? Yes or No.
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Give a reason for your answer.
Straight line LM has equation y = 4x - 7
<br>
Straight line ST has equation y = (9 - x) / 4
<br>
Are the lines LM and ST perpendicular? Yes or No.
<br>
Give a reason for your answer.
How to approach this question
1. Recall the condition for two lines to be perpendicular: the product of their gradients must be -1.
2. Identify the gradient of the first line, LM, from its equation y = 4x - 7.
3. Rearrange the equation for the second line, ST, into the standard form y = mx + c to identify its gradient.
4. Multiply the two gradients together.
5. If the product is -1, the lines are perpendicular. If not, they are not. State your conclusion and show the product of the gradients as your reason.
Full Answer
**Answer: Yes**, the lines are perpendicular..
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**Reason:**
For two lines to be perpendicular, the product of their gradients must be -1 (i.e., m₁ × m₂ = -1).
**Step 1: Find the gradient of line LM.**
The equation y = 4x - 7 is in the form y = mx + c, where m is the gradient.
So, the gradient of LM (m₁) is 4.
**Step 2: Find the gradient of line ST.**
We need to rearrange the equation y = (9 - x) / 4 into the form y = mx + c.
y = 9/4 - x/4
y = -1/4 x + 9/4
So, the gradient of ST (m₂) is -1/4.
**Step 3: Check if the product of the gradients is -1.**
m₁ × m₂ = 4 × (-1/4) = -4/4 = -1.
**Conclusion:**
The product of the gradients is -1. Therefore, the lines **are** perpendicular.
To determine if two lines are perpendicular, we need to compare their gradients. Two lines with gradients m₁ and m₂ are perpendicular if and only if their product is -1 (m₁ × m₂ = -1).
**1. Gradient of line LM:**
The equation is y = 4x - 7. This is in the form y = mx + c, where m is the gradient.
So, the gradient of LM is m₁ = 4.
**2. Gradient of line ST:**
The equation is y = (9 - x) / 4. We need to rewrite this in the form y = mx + c.
y = 9/4 - x/4
y = (-1/4)x + 9/4
So, the gradient of ST is m₂ = -1/4.
**3. Check the product of the gradients:**
m₁ × m₂ = 4 × (-1/4)
= -4/4
= -1
Since the product of the gradients is -1, the lines LM and ST are perpendicular.
So, the answer is **Yes**.
Common mistakes
✗ Incorrectly identifying the gradient of the second line. A common error is to think the gradient is 1/4 or -1, by not correctly separating the fraction.
✗ Stating the rule for parallel lines (gradients are equal) instead of perpendicular lines.
✗ Making a calculation error when multiplying the gradients.
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