ACT Math: Master Substitution Method to Solve Systems of Equations Efficiently

The substitution method works best when one equation is already solved for a variable. Steps: (1) Solve one equation for a variable (if not already done). (2) Substitute that expression into the other equation. (3) Solve for the remaining variable. (4) Substitute back to find the other variable. (5) Verify by plugging both values into both original equations. Example: y=2x and x+y=9. Substitute: x+2x=9, so 3x=9, x=3. Then y=2(3)=6. Check: y=2(3)✓ and 3+6=9✓. Substitution is faster than elimination when one equation is already solved for a variable; recognizing this saves time.

Another example: 3x+y=7 and 2x-y=3. First equation is not solved, so solve: y=7-3x. Substitute: 2x-(7-3x)=3, 2x-7+3x=3, 5x=10, x=2. Then y=7-3(2)=1. Check both equations; both work.

Trap 1: Forgetting to substitute back to find the second variable. After solving for x, you must plug x into an original equation to find y. Trap 2: Making an algebra error when substituting. If y=7-3x and you substitute into 2x-y=3, you get 2x-(7-3x)=3. Many students write 2x-7-3x instead of 2x-7+3x (forgetting to distribute the negative sign), leading to wrong answers. When you substitute, use parentheses: 2x-(7-3x) makes it clear you're subtracting the entire expression, so the negative distributes correctly.

Before you lock in your answer, verify by plugging both variables back into both original equations. This verification step catches algebra errors instantly.

System 1: y=3x and x+y=8. Substitute: x+3x=8, 4x=8, x=2. Then y=3(2)=6. Check: y=3(2)✓ and 2+6=8✓. System 2: 2x+y=5 and x-y=1. From the second: x=y+1. Substitute: 2(y+1)+y=5, 2y+2+y=5, 3y=3, y=1. Then x=1+1=2. Check: 2(2)+1=5✓ and 2-1=1✓. System 3: x=2y-3 and 3x+2y=19. Substitute: 3(2y-3)+2y=19, 6y-9+2y=19, 8y=28, y=3.5. Then x=2(3.5)-3=4. Check: x=2(3.5)-3✓ and 3(4)+2(3.5)=19✓. All three follow the same pattern: solve (or identify) one expression, substitute, solve, verify.

Do ten more system problems using substitution. Write every step including the verification. By test day, substitution will be fast and error-free.

Some systems solve faster with elimination; others solve faster with substitution. Knowing both methods and choosing wisely saves time. If one equation is already solved for a variable, substitution is usually faster; use elimination if coefficients are convenient to eliminate.

This week, practice both elimination and substitution. Learn to recognize which method is faster for each system. By test day, you'll choose the optimal method instantly and solve systems in seconds.