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A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021
Question 3
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. The diagram below shows the line l: 3x - 4... show full transcript
Worked Solution & Example Answer:A line n passes through the points A(-1, 2) and B(0, -2) - Leaving Cert Mathematics - Question 3 - 2021
Step 1
Find the equation of n
Answer
To find the equation of the line passing through points A(-1, 2) and B(0, -2), we first calculate the slope (m) using the formula: Substituting in A(-1, 2) and B(0, -2):
Next, we can use the point-slope form of the equation: y - y_1 = m(x - x_1) Substituting point A(-1, 2): which simplifies to: y - 2 = -4(x + 1) y - 2 = -4x - 4 Therefore: y = -4x - 2.$
Step 2
i) Find the equation of the line k
Answer
To find the equation of line k which is perpendicular to line l, we first find the slope of line l from its equation 3x - 4y = 5. Rearranging to slope-intercept form (y = mx + b):
y = \frac{3}{4}x - \frac{5}{4}$$ The slope of line l is $ rac{3}{4}$, therefore, the perpendicular slope (m_k) is the negative reciprocal: m_k = -\frac{4}{3}. Now, using point P(6, -3): y - (-3) = -\frac{4}{3}(x - 6) Simplifying: y + 3 = -\frac{4}{3}x + 8\ \ y = -\frac{4}{3}x + 5\ Bringing it to the form ax + by + c = 0: \ 4x + 3y - 15 = 0.Step 3
ii) Find the point of intersection of the lines
Answer
To find the point of intersection of lines l: 3x - 4y = 5 and h: 2x - y = 10, we can use substitution or elimination. Let's apply substitution.
From line h: y = 2x - 10. Substituting this into line l:
3x - 8x + 40 = 5 \\ -5x + 40 = 5 \\ -5x = -35 \\ x = 7.$$ Now substituting x back into h: y = 2(7) - 10 = 14 - 10 = 4. The point of intersection is (7, 4).