Question 43 of 49advanced🔧 ApplyDiagram5 marks

Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. Are any of the lines parallel?

(i)

y = -3x + 4

Answer

Slope: -3, y-intercept: 4, cuts y-axis at (0, 4)

(ii)

2y = 4x + 7

Answer

Slope: 2, y-intercept: 7/2, cuts y-axis at (0, 7/2)

(iii)

5y = 6x - 10

Answer

Slope: 6/5, y-intercept: -2, cuts y-axis at (0, -2)

(iv)

3y = 6x - 11

Answer

Slope: 2, y-intercept: -11/3, cuts y-axis at (0, -11/3)

Explanation

To draw the graphs, we first express each equation in the standard form y = ax + b, where a represents the slope and b represents the y-intercept. For equation (i) y = -3x + 4, the slope is -3 and y-intercept is 4. For (ii) 2y = 4x + 7, dividing by 2 gives y = 2x + 7/2, so the slope is 2 and y-intercept is 7/2. For (iii) 5y = 6x - 10, dividing by 5 gives y = (6/5)x - 2, so the slope is 6/5 and y-intercept is -2. For (iv) 3y = 6x - 11, dividing by 3 gives y = 2x - 11/3, so the slope is 2 and y-intercept is -11/3. The coordinates where these lines cut the y-axis are found by setting x = 0, which directly gives the y-intercept points. For (i), the point is (0, 4). For (ii), the point is (0, 7/2). For (iii), the point is (0, -2). For (iv), the point is (0, -11/3).

To draw the graphs, we can plot these y-intercept points and find another point for each line by setting y = 0, then joining the points using a ruler, similar to the examples in the chapter.

Regarding parallel lines, we know that lines with equal slopes but different y-intercepts are parallel to each other. Lines (ii) and (iv) both have a slope of 2, but their y-intercepts (7/2 and -11/3) are different. Therefore, lines (ii) and (iv) are parallel.