Systems by Graphing

Grade 8 · algebra · 100 practice problems · read aloud

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Systems by Graphing 🧮

What is it and Why is it Useful?

A system of equations is two or more equations with the same variables. Solving by graphing means finding the point where the lines cross. This point is the solution that makes both equations true at the same time! It's a visual way to understand how different linear relationships can intersect.

Step-by-Step Guide

  1. Write each equation in slope-intercept form (y = mx + b).
  2. Graph the first equation using the y-intercept (b) and slope (m).
  3. Graph the second equation on the same coordinate plane.
  4. Find the intersection point where the two lines cross.
  5. Check your solution by plugging the (x, y) point into both original equations.

Visual Examples

Example 1: Solve the system: y = 2x + 1 and y = -x + 4

  1. Both are already in y=mx+b form. ✅
  2. Graph y=2x+1: y-intercept at (0,1), slope is 2 (up 2, right 1).
  3. Graph y=-x+4: y-intercept at (0,4), slope is -1 (down 1, right 1).
  4. The lines intersect at (1, 3).
  5. Check: 3 = 2(1)+1? Yes! 3 = -(1)+4? Yes! ✔️

Example 2: Solve the system: y = 3x - 2 and y = 3x + 1

  1. Both lines have the same slope (m=3) but different y-intercepts.
  2. When graphed, these lines are parallel and never intersect.
  3. This means there is no solution to this system. ❌

Common Mistakes to Avoid

  • Incorrect Slope: Mixing up rise over run. Remember: slope = change in y / change in x.
  • Poor Graphing: Not using a ruler or being careless can lead to an inaccurate intersection point. Be neat!
  • Forgetting to Check: Always verify your solution in both original equations.
  • One Solution Only: Assuming all systems have one answer. Remember: some have none (parallel lines) or infinitely many (same line).

Tips & Tricks

  • 👉 Use graph paper for accuracy.
  • 👉 If the intersection point isn't a nice whole number, your graphing might be slightly off—double-check your work!
  • 👉 Remember the three possible outcomes: One Solution (lines cross), No Solution (parallel lines), Infinite Solutions (same line).

How to Practice

Start with systems that are already in slope-intercept form. Then, move on to problems where you have to rearrange the equations first. You can find practice worksheets online or create your own! Try solving at least 3-5 problems a day to build confidence. ✏️

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 2x + 3y = 12; x - y = 1

Hint: To solve by graphing, first rearrange each equation into slope-intercept form. Then plot both lines on the same coordinate plane and identify where they intersect.

Show the answer

Answer: (3, 2)

  1. Solve one equation for one variable. From Equation (2): x - y = 1 Add y to both sides: x = 1 + y
  2. Substitute this expression for x into Equation (1). Equation (1) is: 2x + 3y = 12 Replace x with (1 + y): 2(1 + y) + 3y = 12
  3. Simplify and solve for y. 2 * 1 + 2 * y + 3y = 12 2 + 2y + 3y = 12 2 + 5y = 12 Subtract 2 from both sides: 5y = 10 Divide both sides by 5: y = 2
  4. Substitute y = 2 back into the expression for x. From
  5. x = 1 + y x = 1 + 2 x = 3
  6. Write the solution as an ordered pair (x, y). The solution is (3, 2)
  7. Check the solution in both original equations. Check Equation (1): 2(3) + 3(2) = 6 + 6 = 12 ✓ Check Equation (2): 3 - 2 = 1 ✓ Both equations are satisfied, so the solution is correct.

We are solving the system of equations: Equation (1): 2x + 3y = 12 Equation (2): x - y = 1

2 2x + y = 7 and y = x - 1

Hint: Substitute the expression for y from the second equation into the first equation to solve for x, then find y.

Show the answer

Answer: (2.67, 1.67)

  1. Substitute y = x - 1 into the first equation: 2x + (x - 1) = 7
  2. Combine like terms: 3x - 1 = 7
  3. Add 1 to both sides: 3x = 8
  4. Divide both sides by 3: x = 8/3 ≈ 2.67
  5. Substitute x = 8/3 into y = x - 1: y = 8/3 - 1 = 8/3 - 3/3 = 5/3 ≈ 1.67
  6. The solution is (8/3, 5/3) or approximately (2.67, 1.67)

3 y = 2x - 3 and y = -x + 6

Hint: To solve by graphing, find where the two lines intersect by plotting both equations on the same coordinate plane.

Show the answer

Answer: (3, 3)

  1. Set the equations equal to find the x-coordinate: 2x - 3 = -x + 6
  2. Add x to both sides: 3x - 3 = 6
  3. Add 3 to both sides: 3x = 9
  4. Divide by 3: x = 3
  5. Substitute x = 3 into either equation to find y: y = 2(3) - 3 = 6 - 3 = 3
  6. The solution is the ordered pair (3, 3)

The answer is (3, 3).

4 2x + y = 8 and y = -x + 5

Hint: Remember that the solution is the point where both lines intersect on a graph. You can find this by substituting one equation into the other.

Show the answer

Answer: (1, 6)

  1. Substitute the expression for y from the second equation into the first equation. 2x + (-x + 5) = 8
  2. Simplify and solve for x. 2x - x + 5 = 8 x + 5 = 8 x = 8 - 5 x = 3
  3. Substitute x = 3 back into the second equation to find y. y = -3 + 5 y = 2
  4. The solution is the ordered pair (x, y). The solution is (3, 2).

5 y = 2x - 1 and y = -x + 5

Hint: Find where the two lines intersect by setting the expressions for y equal to each other.

Show the answer

Answer: (2, 3)

  1. Set the two expressions for y equal to each other: 2x - 1 = -x + 5
  2. Add x to both sides: 3x - 1 = 5
  3. Add 1 to both sides: 3x = 6
  4. Divide both sides by 3: x = 2
  5. Substitute x = 2 into either equation to find y: y = 2(2) - 1 = 4 - 1 = 3
  6. The solution is the ordered pair (2, 3).

6 y = 6x - 1 and y = -x + 6

Hint: Graph both lines on the same coordinate plane. The solution is the point where the two lines cross. You can also set the equations equal to each other to find the x-coordinate of the intersection.

Show the answer

Answer: (1, 5)

  1. Set the equations equal to find x: 6x - 1 = -x + 6
  2. Add x to both sides: 7x - 1 = 6
  3. Add 1 to both sides: 7x = 7
  4. Divide both sides by 7: x = 1
  5. Substitute x = 1 into y = 6x - 1: y = 6(1) - 1 = 6 - 1 = 5
  6. Verify by substituting into the other equation: y = -1 + 6 = 5 ✓
  7. The solution is the intersection point: (1, 5)
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