Solving Systems by Elimination

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Solving Systems by Elimination: Making Variables Disappear

Imagine you're at a carnival game where you need to find the price of one ring toss and one dart throw. The booth owner gives you two clues: "3 ring tosses + 2 dart throws = letter: 'R', title: 'Solving Systems by Elimination', concept: 1" and "3 ring tosses + 4 dart throws = letter: 'R', title: 'Solving Systems by Elimination', concept: 7." How do you solve this puzzle when you have two unknowns?

This is exactly what a system of equations represents — multiple equations working together to find unknown values. The elimination method is like a magic trick: we strategically add equations together to make one variable completely disappear.

The Power of Opposite Coefficients

The elimination method works best when variables have opposite coefficients — like +3x and -3x, or +5y and -5y. When you add opposites, they cancel out to zero.

Let's solve our carnival problem step by step:

Step-by-Step Solution:

Given:
Equation 1: 3r + 2d = 11
Equation 2: 3r + 4d = 17

Step 1: Notice both equations have +3r. To eliminate r, we need opposite coefficients. Multiply Equation 1 by -1:
-3r - 2d = -11

Step 2: Add the modified equation to Equation 2:
(-3r - 2d) + (3r + 4d) = -11 + 17
0r + 2d = 6
2d = 6
d = 3

Step 3: Substitute d = 3 back into either original equation:
3r + 2(3) = 11
3r + 6 = 11
3r = 5
r = 5/3

💡 The "Cancellation Magic"

Here's the counterintuitive part: adding equations together actually helps us solve them. It seems like we're making things more complicated, but we're strategically simplifying.

When 3r meets -3r, they don't just disappear — they've done their job by revealing what the other variable must be. It's like removing one puzzle piece to see the whole picture clearly.

When Coefficients Aren't Already Opposites

Sometimes you need to create opposite coefficients by multiplying one or both equations by strategic numbers. If you have 2x and 3x, you could multiply the first equation by 3 and the second by -2 to get 6x and -6x — perfect opposites ready for elimination.

🔑 Key Takeaway

Just like our carnival puzzle, elimination method turns two mysterious relationships into one clear answer. By strategically making variables cancel out, we transform the complexity of two unknowns into the simplicity of one equation, one unknown — and suddenly the impossible becomes inevitable.

Sample questions

1. Solve by elimination: x + y = 8 and x - y = 2

✓ (5, 3)

○ (3, 5)

○ (4, 4)

○ (6, 2)

Answer: (5, 3) — Add: (x+y)+(x-y)=8+2 → 2x=10 → x=5, then 5+y=8 → y=3.

2. Solve: 2x + y = 7 and 3x - y = 8

○ (2, 3)

✓ (3, 1)

○ (4, -1)

○ (1, 5)

Answer: (3, 1) — Add: (2x+y)+(3x-y)=7+8 → 5x=15 → x=3, then 2(3)+y=7 → 6+y=7 → y=1.

3. What is the result when adding x + 2y = 10 and x - 2y = 2?

○ 2y = 12 → y = 6

○ 2x = 8 → x = 4

✓ 2x = 12 → x = 6

○ 2y = 8 → y = 4

Answer: 2x = 12 → x = 6 — x + x = 2x, 2y + (-2y) = 0, 10+2=12 → 2x=12 → x=6.

Skills in this topic

Solve a system of equations by addition (eliminating a variable with opposite coefficients)
Solve a system of equations by subtraction (eliminating a variable with identical coefficients)
Solve a system by multiplying one equation by a constant to eliminate a variable
Solve a system by multiplying both equations by different constants to eliminate a variable
Choose the most efficient method (graphing, substitution, or elimination) to solve a given system

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