Grade 8: Simultaneous Linear Equations

Objective

Students will understand that the solution to a pair of simultaneous linear equations is the point (x, y) that makes both equations true at the same time, and that this solution corresponds to the point where the graphs of the two lines intersect.

Materials

• "Grade 8: Simultaneous Linear Equations" resource (Chapter 4, Section 4.1 — Examples 1–7 and Summary)
• Graph paper (or copies of Figures 1–4 if printable)
• Pencils
• Whiteboard or chart paper for warm-up problem

Warm-up (~5 min)

Write this on the board and have students work silently, then share answers:

"I am thinking of two numbers. Their sum is 38. Their difference is 4. What are the two numbers?"

• Give students 2–3 minutes to guess-and-check.
• Ask for answers (x = 21, y = 17).
• Tell students: "Today we'll learn the math name for this kind of problem — simultaneous linear equations — and two ways to solve them: algebra and graphing."

Main Activity (~25 min)

Part 1: Algebraic approach (~12 min)

1. Read aloud (or have a student read) Example 1 from the resource — the "ages of siblings" problem. Show how "sum is 38" and "difference is 4" become the equations x + y = 38 and x − y = 4.
2. Walk through the addition/subtraction process from the text:
3. Add the equations: (x + y) + (x − y) = 38 + 4 → 2x = 42 → x = 21
4. Subtract the equations: (x + y) − (x − y) = 38 − 4 → 2y = 34 → y = 17
5. Have students copy this worked example into their notes.
6. Read Example 3 (Lovasz and Tonio marbles) as a second example, this time using substitution:
7. L = 5 + 2T and L + T = 107
8. Substitute to get 5 + 2T + T = 107, solve for T = 34, then L = 73.
9. Point out the three tools named in the text:
10. a. Add equals to equals
11. b. Subtract equals from equals
12. c. Multiply equals by a nonzero number
13. d. Substitution (replace an expression with an equal expression from the other equation)

Part 2: Graphical approach (~13 min)

1. Introduce Example 4 from the resource: 3x + y = 7 and x + 3y = 5.
2. Have students graph both lines on graph paper (or refer to Figure 1 in the resource if available).
3. Ask students to identify the point of intersection: (2, 1).
4. Check together that x = 2, y = 1 makes both equations true:
5. 3(2) + 1 = 7 ✓
6. 2 + 3(1) = 5 ✓
7. Briefly review the Summary box from the resource (three cases):
8. Different slopes → lines intersect → one solution
9. Same slope, different intercepts → parallel lines → no solution
10. Same slope, same intercept → same line → infinitely many solutions

Wrap-up / Exit Ticket (~10 min)

Have students answer the following on paper to turn in:

1. In your own words, what does it mean for two equations to be "simultaneous"?
2. Solve using addition or subtraction: x + y = 10 and x − y = 2. What are x and y?
3. If two lines have the same slope but different y-intercepts, how many solutions does the system have? Why?

Collect exit tickets as a check for understanding.

If Time Remains

Have students try Example 5 from the resource on their own graph paper: x − 2y = 8 and 2x + 5y = 34. They should graph both lines, find the point of intersection (12, 2), and check the solution by substituting into both equations. Students who finish early can compare their graph to Figure 2 in the resource.