Solving By Substitution Method

Master the algebraic substitution method to solve linear equations exactly.

9 lessons

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Concept

Why Use Algebra?

The limitation of graphing non-integral coordinates.

Graphing linear equations is visual and intuitive, but it is not always perfectly precise. When the lines intersect at non-integral coordinates like $\sqrt{3}$ or fractions like $\frac{49}{29}$ , reading the exact point directly from a graph is prone to errors.

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The Substitution Steps

The 3 core steps of substitution.

Step 1

Pick one equation and express one variable in terms of the other. For example, write

x

in terms of

y

Step 2

Substitute this expression into the other equation. Then solve for the single remaining variable.

Step 3

Put the value you found back into the Step 1 expression to find the second variable.

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Worked Example

Standard Substitution

Solve a straightforward system by substitution.

Problem. Solve the following pair of equations by the substitution method:

$7x - 15y = 2 \quad \text{(Equation 1)}$ $x + 2y = 3 \quad \text{(Equation 2)}$

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Worked Example

Infinite Solutions Case

Substitution where variables cancel leaving a true statement.

Problem. Find the cost of a pencil and an eraser using the following pair of equations:

$2x + 3y = 9 \quad \text{(Equation 1)}$ $4x + 6y = 18 \quad \text{(Equation 2)}$

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Worksheet

No Solution Case

Substitution where variables cancel leaving a false statement.

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Consider the system of linear equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ . We express $x$ in terms of $y$ from the first equation to get $x = 4 - 2y$ . Now, we substitute this value of $x$ into the second equation to get . Expanding this equation yields $8 - 4y + 4y - 12 = 0$ . After combining like terms, we arrive at the equation , which is a false statement. Because the variables are eliminated and the resulting statement is false, the equations do not have a . Consequently, the rails represented by these lines are parallel and will never cross.

← previousGraphical Method And Line Behavior next →Solving By Elimination Method

Concept

Why Use Algebra?

The limitation of graphing non-integral coordinates.

1 / 3

Worked Example

Standard Substitution

Solve a straightforward system by substitution.

Problem. Solve the following pair of equations by the substitution method:

$7x - 15y = 2 \quad \text{(Equation 1)}$ $x + 2y = 3 \quad \text{(Equation 2)}$

1 / 5

Worked Example

Infinite Solutions Case

Substitution where variables cancel leaving a true statement.

Problem. Find the cost of a pencil and an eraser using the following pair of equations:

$2x + 3y = 9 \quad \text{(Equation 1)}$ $4x + 6y = 18 \quad \text{(Equation 2)}$

1 / 4

Worksheet

No Solution Case

Substitution where variables cancel leaving a false statement.

0 of 3 blanks filled