a) A line n passes through the points A(-1, 2) and B(0, -2). Write the equation of n in the form y = mx + c, where m, c ∈ ℤ. b) The diagram below shows the line l: 3x - 4y = 5 and the point P(6, -3). (i) Find the equation of the line k through the point P that is perpendicular to the line l. Write your answer in the form ax + by + c = 0, where a, b, c ∈ ℤ. (ii) Find the point of intersection of the lines l: 3x - 4y = 5 and h: 2x - y = 10.

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a) A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021

Question 3

a) A line n passes through the points A(-1, 2) and B(0, -2). Write the equation of n in the form y = mx + c, where m, c ∈ ℤ. b) The diagram below shows the line l: ... show full transcript

Worked Solution & Example Answer:a) A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021

Step 1

Write the equation of n in the form y = mx + c

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Answer

To find the equation of the line n, we first need to calculate the slope (m) using the two points A(-1, 2) and B(0, -2).

The formula for the slope (m) is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the coordinates: $m = \frac{-2 - 2}{0 - (-1)} = \frac{-4}{1} = -4$

Now that we have the slope, we can use point-slope form to find the equation: Using point B(0, -2): $y - y_1 = m(x - x_1)$ Substituting the values: $y - (-2) = -4(x - 0)$ This simplifies to: $y + 2 = -4x$ Rearranging gives us: $y = -4x - 2$ Thus, the final equation is: $y = -4x - 2$

Step 2

Find the equation of the line k through the point P that is perpendicular to the line l.

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Answer

First, we need to determine the slope of line l: 3x - 4y = 5. Rewriting this in slope-intercept form gives:

$4y = 3x - 5 \Rightarrow y = \frac{3}{4}x - \frac{5}{4}$

Thus, the slope of line l, m_l = \frac{3}{4}. The slope of line k, which is perpendicular to line l, is the negative reciprocal:

$m_k = -\frac{4}{3}$

Next, using point P(6, -3) and the point-slope formula: $y - y_1 = m(x - x_1)$ Substituting: $y - (-3) = -\frac{4}{3}(x - 6)$

This simplifies to: $y + 3 = -\frac{4}{3}x + 8$ Rearranging leads to: $y = -\frac{4}{3}x + 5$ Now, to convert this to the form ax + by + c = 0:

Multiply through by 3 to eliminate the fraction: $3y = -4x + 15$ Rearranging gives: $4x + 3y - 15 = 0$

Step 3

Find the point of intersection of the lines l: 3x - 4y = 5 and h: 2x - y = 10.

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Answer

To find the point of intersection, we need to solve the system of equations:

3x - 4y = 5
2x - y = 10

From the second equation, express y in terms of x: $y = 2x - 10$

Now substitute for y in the first equation: $3x - 4(2x - 10) = 5$ Expanding this gives: $3x - 8x + 40 = 5$ Combining like terms produces: $-5x + 40 = 5$ Solving for x: $-5x = 5 - 40 \Rightarrow -5x = -35 \Rightarrow x = 7$

Now substituting x = 7 back into the equation for y: $y = 2(7) - 10 = 14 - 10 = 4$

Thus, the point of intersection is (7, 4).

a) A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021

Worked Solution & Example Answer:a) A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021

Write the equation of n in the form y = mx + c

Find the equation of the line k through the point P that is perpendicular to the line l.

Find the point of intersection of the lines l: 3x - 4y = 5 and h: 2x - y = 10.

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