Question

One linear equation is defined by points (1, 6) and (3, 7), while the other is defined by points (3, 6) and (5, 8). Which point represents the solution of this system of equations?

Answer 1

The solution to the system of linear equations represented by two lines passing through points (1, 6) and (3, 7), and (3, 6) and (5, 8) is found to be the point (5, 8).

Step-by-step explanation:

To find the solution of the system of equations represented by two lines, we first need to determine the equations of the lines defined by the given points. Let's find the equations for each line using the point-slope form, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

• For the first line passing through (1, 6) and (3, 7), we calculate the slope as m = (7 - 6) / (3 - 1) = 1 / 2. The equation becomes y - 6 = 1/2(x - 1) or y = 1/2x + 11/2 after rearranging.
• For the second line passing through (3, 6) and (5, 8), the slope is m = (8 - 6) / (5 - 3) = 1. The equation is y - 6 = 1(x - 3) or y = x + 3 after rearranging.

To find the intersection point, which is the solution to the system of linear equations, we set the two equations equal to each other: 1/2x + 11/2 = x + 3. Solving for x gives:

1. Subtract 1/2x from both sides: 11/2 = 1/2x + 3.
2. Subtract 3 from both sides: 5/2 = 1/2x.
3. Multiply both sides by 2: x = 5.

Plugging x = 5 into either of the original equations to solve for y, we take the second equation y = x + 3:

• y = 5 + 3
• y = 8

Therefore, the points (5, 8) represent the solution of this system of linear equations.

Answer 2

Find where they intersect on a coordinate plane.