Solve two linear equations in two variables using elimination and substitution methods, and apply to real-world problems.
In elimination, one variable is removed by adding or subtracting the equations after making its coefficients equal (or opposite).
Solve: x+y=10, x−y=4
Adding: 2x=14, x=7; then y=3
Solve: 2x+3y=13, 2x−y=1
Subtracting: 4y=12, y=3; then 2x=1+3=4, x=2
Solve: 2x-y=3, 3x+y=17
Adding: 5x=20, x=4; then y=5
In substitution, one equation is solved for one variable, then substituted into the other equation.
Solve using substitution: x=y+2, 2x+y=13
2(y+2)+y=13 → 3y+4=13 → y=3, x=5
The sum of two numbers is 15 and their difference is 5. Find the numbers.
x+y=15, x−y=5; adding: 2x=20, x=10; y=5
Solve using substitution: y=3x-1, 2x+y=14
2x+3x-1=14 → 5x=15 → x=3, y=8
1. Solve: x+y=12, x−y=2
2. Solve: 3x+2y=16, 3x−y=7
3. Solve using substitution: y=2x+1, x+y=13
4. The sum of two numbers is 20 and their difference is 8. Find the numbers.
5. Solve: 4x+y=14, x+y=5
6. Solve: x+2y=8, x-y=2
7. Solve: 4x-y=10, 2x+y=8
8. Solve using substitution: y=x+4, 3x+y=16
9. The sum of two numbers is 25 and their difference is 7. Find the numbers.
10. Solve: 5x+2y=19, x-y=1