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A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.
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A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.
Both lines have a slope of , and the y-intercepts are different. So the lines are parallel. Example 3 Determine whether the lines are parallel, intersect, or coincide. A. y = 3x + 7, y = –3x – 4 The lines have different slopes, so they intersect. B. Solve the second equation for y to find the slope-intercept form.
C. 2y – 4x = 16, y – 10 = 2(x - 1) Solve both equations for y to find the slope-intercept form. 2y – 4x = 16 y – 10 = 2(x – 1) 2y = 4x + 16 y – 10 = 2x - 2 y = 2x + 8 y = 2x + 8 Both lines have a slope of 2 and a y-intercept of 8, so they coincide.
Example 4: Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same? The answer is the number of miles for which the costs of the two plans would be the same. Plan A costs $100.00 for the initial fee and $0.35 per mile. Plan B costs $85.00 for the initial fee and $0.50 per mile.
0 = –0.15x + 15 Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations. Plan A: y = 0.35x + 100 Plan B: y = 0.50x + 85 Subtract the second equation from the first. x = 100 Solve for x. Substitute 100 for x in the first equation. y = 0.50(100) + 85 = 135
The lines cross at (100, 135). Both plans cost $135 for 100 miles. What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan? The lines would be parallel.
