Pair of Linear Equations in Two Variables · Lesson 2 of 4
Algebraic Method 1: The Substitution Method
“It’s essentially mathematical identity theft where you isolate one variable just to force its partner to do all the heavy lifting in the next equation.”
• Explain why algebraic methods are useful when graphical solutions are difficult to read accurately. • Describe the substitution method for solving a pair of linear equations. • Express one variable in terms of the other variable. • Substitute the resulting expression into the second equation. • Find the values of both variables and verify the final solution.
Introduction
The graphical method gives a useful visual representation of a pair of linear equations. However, it may become inconvenient when the solution contains fractions, decimals or irrational values. In such cases, reading the exact coordinates from a graph can be difficult.
Algebraic methods avoid graphical estimation and use exact calculations. The substitution method works by converting a pair of equations in two variables into a single equation containing only one variable.
Express one variable in terms of the other variable, substitute that expression into the second equation, and then solve the resulting single-variable equation.
Problem
Standard Step-by-Step Procedure
- 1.Express one variable in terms of the other: Choose one equation and isolate either x or y. Prefer the variable that is easiest to isolate.
- 2.Substitute into the other equation: Replace the isolated variable in the second equation with the expression obtained in Step 1. This produces an equation in only one variable.
- 3.Solve for the remaining variable: Simplify the single-variable equation and calculate its numerical value.
- 4.Find the other variable: Substitute the value found in Step 3 into the expression obtained in Step 1.
- 5.Write and verify the solution: Write the answer as an ordered pair and check that it satisfies both original equations.
Complete Worked Example
Problem
Solve the given pair of linear equations using substitution.
- 1.Choose the equation in which one variable can be isolated most easily.
- 2.Isolate x from the second equation.
- 3.Substitute the resulting expression into the first equation.
- 4.Solve the resulting equation for y.
- 5.Substitute the value of y back into the expression for x.
- 6.Write the final ordered pair.
Isolate One Variable
Equation (2) is the easiest equation to rearrange because the coefficient of x is 1. Isolate x.
Substitute into the Other Equation
Replace x in Equation (1) with the expression 3 − 2y.
Distribute 7 across the terms inside the parentheses.
Combine the like terms containing y.
Solve for y
The first variable value obtained is y = 19/29.
Substitute Back to Find x
Substitute the value of y into Equation (3), where x = 3 − 2y.
Write 3 with denominator 29 and simplify.
Verify the Solution
Substitute the values of x and y into both original equations to confirm that they satisfy the system.
The values satisfy both original equations. Therefore, the pair has one unique solution.
When using substitution, first look for a variable whose coefficient is 1 or −1. Such a variable can usually be isolated without introducing fractions at the beginning of the solution.
Use parentheses when substituting an expression, distribute coefficients to every term inside the parentheses, combine signs carefully, and always substitute back to find the second variable.
Quiz
What is the first main operation in the substitution method?
Which expression for x follows from x + 2y = 3?
Why is a variable with coefficient 1 or −1 usually convenient to isolate?
How should a proposed solution to a pair of linear equations be verified?
For 2x + y = 11 and x − y = 1, which single-variable equation results after using x = y + 1?
Practice Problems
- Solve x + y = 9 and x − y = 3 using substitution.
- Solve 2x + y = 11 and x − y = 1 using substitution.
- Solve 3x + 2y = 16 and x = 2y using substitution.
- Solve 4x − 3y = 11 and 2x + y = 9 using substitution. Express fractional answers in simplest form.
- The sum of two numbers is 27. The first number is 3 more than twice the second number. Form a pair of equations and solve it using substitution.
Key Takeaways
• The substitution method reduces a pair of two-variable equations to one equation in one variable. • Choose a variable that is easy to isolate, especially one with coefficient 1 or −1. • Use parentheses when replacing a variable with an expression. • Distribute coefficients to every term and combine like terms carefully. • Substitute the first calculated value back to find the other variable. • Write the solution as an ordered pair and verify it in both original equations.