Solving by Substitution

Solving Systems by Substitution

A system of equations is a set of two (or more) equations that share the same variables. Solving the system means finding the value(s) that make all the equations true at the same time — geometrically, this is the point where the lines intersect.

The Substitution Method

Substitution works best when one equation is already solved for a variable (or is easy to solve for one). The idea: solve one equation for one variable, then substitute that expression into the other equation, which leaves you with a single equation in a single variable.

1. Solve one of the equations for one variable in terms of the other.
2. Substitute that expression into the other equation, replacing the variable you solved for.
3. Solve the resulting one-variable equation.
4. Substitute that value back into either original equation to find the other variable.
5. Write the answer as an ordered pair (x, y) and check it in both original equations.

Example

Solve the system:

1. The first equation is already solved for y: y = 2x + 1.
2. Substitute 2x + 1 in place of y in the second equation: 3x + (2x + 1) = 16
3. Combine like terms and solve for x: 5x + 1 = 16 \;\Rightarrow\; 5x = 15 \;\Rightarrow\; x = 3
4. Substitute x = 3 back into y = 2x + 1: y = 2(3) + 1 = 7
5. The solution is (x, y) = (3, 7).

Checking the Solution

Substitute (3, 7) into both original equations:

• y = 2x+1: 7 = 2(3)+1 = 7. True.
• 3x+y=16: 3(3)+7 = 9+7 = 16. True.

Both check out, confirming (3, 7) is the correct solution.

When Neither Equation Is Pre-Solved

If neither equation is already solved for a variable, look for the easiest one to isolate — usually a variable with a coefficient of 1. For the system x + 2y = 9 and 3x - y = 8, the x in the first equation is easiest to isolate: x = 9 - 2y. Substituting into the second equation gives 3(9 - 2y) - y = 8, which becomes 27 - 6y - y = 8, or 27 - 7y = 8. Solving gives y = \tfrac{19}{7} — showing that substitution still works even when the numbers aren't as clean, though it's a good sign to double-check your arithmetic whenever fractions like this appear.

Key Takeaways

• Substitution works best when a variable is already isolated, or easy to isolate.
• Substitute the expression into the other equation, not the one you solved.
• Always report the solution as an ordered pair and check it in both original equations.