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Algebra Linear Function Example 2

Question

Line1 pass the point A and intersects with line2 at point B. The equation of line2 is y = x/2. The point A lies on y-axis and its value is 3. The x-coordinate of the point B is 2. What is the equation of line1?

The graph shows that the line l1 intersects the y-axis at 3 and two lines intersect at point B whose x-coordinate is 3.

Solution:

The equation of line1 is y = k x + b, in which k is the slope of the line and b is the y-intercept of the line. We need to determine the coefficient k and b.

Look for point B. Point B lies on both line1 and line2 and the x-coordinate of the point B is 2. We are given the equation of line2. Because any point on a line will satisfy its equation. So, substitute the coordinate of the point B into its equation, we get the y-coordinate of point B.

substitute B(2, y) into equation y = x/2
y = 2/2 = 1
So, the coordinate of the point B is (2, 1).

Because point A and point B lie on line1, so point A and point B satisfy the equation of line1. Substitute points A (0, 3) and B (2, 1) into the line1 equation y = k x + b to get the coefficient k and b.

Substitute A (0, 3) into the line1 equation y = k x + b
3 = k × 0 + b
b = 3.
Substitute B (2, 1) into the line1 equation y = k x + 3
1 = k × 2 + 3
2 k = 1 - 3 = -2
k = -1

Substitute b = 3 and k = -1 into the line equation y = k x + b, we get the equation of line1 which is y = -x + 3.