Guided Practice
Overview
This lesson provides structured practice solving systems of linear equations using both graphing and substitution methods. Work through each problem step-by-step, checking your understanding as you go.
Practice Problems
Question 1: Solve the system by graphing: y = 2x + 1 and y = -x + 4
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Answer: (1, 3)
The lines intersect at x = 1, y = 3. Check: 3 = 2(1) + 1 = 3 and 3 = -1 + 4 = 3.
Question 2: Solve by substitution: y = 3x and 2x + y = 15
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Answer: (3, 9)
Substitute y = 3x into 2x + y = 15: 2x + 3x = 15, 5x = 15, x = 3. Then y = 3(3) = 9.
Question 3: Solve the system: x + y = 10 and x - y = 2
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Answer: (6, 4)
From x - y = 2, we get x = y + 2. Substitute: (y + 2) + y = 10, 2y = 8, y = 4. Then x = 6.
Question 4: Solve the system: 3x + 2y = 12 and y = x - 1
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Answer: (2.8, 1.8) or (14/5, 9/5)
Substitute y = x - 1: 3x + 2(x - 1) = 12, 3x + 2x - 2 = 12, 5x = 14, x = 14/5 = 2.8. Then y = 2.8 - 1 = 1.8.
Question 5: Solve by graphing: y = x + 2 and y = x - 3
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Answer: No solution (parallel lines)
Both equations have slope 1 but different y-intercepts, so they are parallel and never intersect.
Question 6: Solve the system: 2x + 3y = 13 and x = 2
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Answer: (2, 3)
Substitute x = 2: 2(2) + 3y = 13, 4 + 3y = 13, 3y = 9, y = 3.
Question 7: Solve the system: y = 4x - 5 and 2y = 8x - 10
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Answer: Infinitely many solutions
The second equation simplifies to y = 4x - 5, which is the same as the first equation. The lines are identical.
Question 8: Solve: x + 2y = 8 and 3x - y = 3
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Answer: (2, 3)
From x + 2y = 8, x = 8 - 2y. Substitute: 3(8 - 2y) - y = 3, 24 - 6y - y = 3, -7y = -21, y = 3. Then x = 8 - 6 = 2.
Question 9: Solve: y = -2x + 7 and y = 3x - 8
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Answer: (3, 1)
Set equal: -2x + 7 = 3x - 8, 15 = 5x, x = 3. Then y = -2(3) + 7 = 1.
Question 10: Solve: 4x - y = 10 and 2x + y = 8
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Answer: (3, 2)
From 2x + y = 8, y = 8 - 2x. Substitute: 4x - (8 - 2x) = 10, 6x - 8 = 10, 6x = 18, x = 3. Then y = 8 - 6 = 2.