Coefficients Already Opposites

Solve the system: $$\begin{cases} x + 2y = 7 \\ x - 2y = 1 \end{cases}$$

Identify opposite coefficients: The $y$ coefficients are $+2$ and $-2$ - they're opposites! = Ready to add

Add the equations: $(x + 2y) + (x - 2y) = 7 + 1$ $2x + 0y = 8$ = $2x = 8$

Solve for $x$: $2x = 8$ $x = 4$ = $x = 4$

Substitute back to find $y$: $4 + 2y = 7$ $2y = 3$ $y = 1.5$ = $y = 1.5$

Verify the solution: Equation 1: $4 + 2(1.5) = 4 + 3 = 7$ ✓ Equation 2: $4 - 2(1.5) = 4 - 3 = 1$ ✓ = Both equations satisfied

Answer: The solution is $(4, 1.5)$

Multiply to Create Opposites

Solve the system: $$\begin{cases} 3x + 2y = 16 \\ x + y = 6 \end{cases}$$

Choose a variable to eliminate: Let's eliminate $y$. We need the $y$ coefficients to be opposites. = Target: eliminate $y$

Multiply equation 2 by $-2$: $-2(x + y) = -2(6)$ $-2x - 2y = -12$ = New equation 2: $-2x - 2y = -12$

Add the equations: $(3x + 2y) + (-2x - 2y) = 16 + (-12)$ $x + 0y = 4$ = $x = 4$

Substitute back to find $y$: $4 + y = 6$ $y = 2$ = $y = 2$

Verify the solution: Equation 1: $3(4) + 2(2) = 12 + 4 = 16$ ✓ Equation 2: $4 + 2 = 6$ ✓ = Both equations satisfied

Answer: The solution is $(4, 2)$

Multiply Both Equations

Solve the system: $$\begin{cases} 2x + 3y = 13 \\ 3x + 2y = 12 \end{cases}$$

Choose a variable to eliminate: Let's eliminate $x$. Current coefficients: $2$ and $3$. LCM is $6$. = Need coefficients $6$ and $-6$

Multiply equation 1 by $3$: $3(2x + 3y) = 3(13)$ $6x + 9y = 39$ = $6x + 9y = 39$

Multiply equation 2 by $-2$: $-2(3x + 2y) = -2(12)$ $-6x - 4y = -24$ = $-6x - 4y = -24$

Add the equations: $(6x + 9y) + (-6x - 4y) = 39 + (-24)$ $5y = 15$ = $y = 3$

Substitute back to find $x$: $2x + 3(3) = 13$ $2x + 9 = 13$ $2x = 4$ $x = 2$ = $x = 2$

Verify the solution: Equation 1: $2(2) + 3(3) = 4 + 9 = 13$ ✓ Equation 2: $3(2) + 2(3) = 6 + 6 = 12$ ✓ = Both equations satisfied

Answer: The solution is $(2, 3)$

Mistake: Forgetting to multiply the entire equation

Why: When multiplying to create opposite coefficients, every term must be multiplied, including the constant on the right side.

Correct: If multiplying $x + 2y = 5$ by $3$, you get $3x + 6y = 15$, not $3x + 6y = 5$.

Mistake: Adding when you should subtract (or vice versa)

Why: For elimination to work, the coefficients must be true opposites (like $+3$ and $-3$). If they're the same, you need to subtract instead.

Correct: Same signs: subtract the equations. Opposite signs: add the equations.

Mistake: Not verifying the solution in both equations

Why: A solution must satisfy both original equations. Checking only one equation can miss errors.

Correct: Always substitute your answer into BOTH original equations to verify.

Concert Ticket Sales

Event managers use systems of equations to analyze ticket sales when different types have different prices.

A concert sold 500 tickets total. VIP tickets cost 80 euros and regular tickets cost 40 euros. Total revenue was 28,000 euros. How many of each type were sold?

Chemistry: Balancing Mixtures

Chemists use elimination to determine how much of each solution to mix to achieve a desired concentration.

A chemist needs 100 mL of a 30% acid solution. She has 20% and 50% acid solutions available. How much of each should she mix?

Q: When should I use elimination instead of substitution?

A: Use elimination when: (1) the coefficients are already opposites or equal, (2) neither equation is already solved for a variable, or (3) substitution would create messy fractions. Use substitution when one equation is already solved for a variable (like $y = 3x + 2$).

Q: What if both variables get eliminated?

A: If you get a true statement like $0 = 0$, the system has infinitely many solutions (the lines are the same). If you get a false statement like $0 = 5$, the system has no solution (the lines are parallel).

Q: Does it matter which variable I eliminate first?

A: No! You'll get the same answer either way. Choose the variable that requires the simplest multiplication to create opposite coefficients.