Solving By Elimination Method

Master the elimination method by equalizing coefficients and adding/subtracting equations.

9 lessons

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Why Elimination?

Introduce elimination as a more convenient alternative to substitution.

While substitution works for every pair of linear equations, it can often lead to messy fractions and long calculations. The elimination method offers a faster, cleaner alternative by completely removing a variable in one swift step.

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The Elimination Steps

The 4 core steps of the elimination method.

Step 1

Multiply

Multiply equations by suitable constants so that one variable has equal coefficients.

Step 2

Eliminate

Add or subtract the equations to remove that variable completely.

Step 3

Solve

Solve the new single-variable equation to find the value of one variable.

Step 4

Substitute

Put the solved value back into either original equation to find the second variable.

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Special Cases in Elimination

What happens when variables disappear entirely?

True Statement

If elimination gives a true statement with no variables, such as $9 = 9$ , the two lines are coincident.

Infinitely many solutions

False Statement

If elimination gives a false statement, such as $0 = 9$ , the two lines are parallel.

No solution and inconsistent pair

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Eliminating 'y'

Solve a word problem using elimination.

Problem. The ratio of incomes of two persons is $9 : 7$ and the ratio of their expenditures is $4 : 3$ . If each of them manages to save ₹ 2000 per month, find their monthly incomes.

Let the incomes be $9x$ and $7x$ . Let expenditures be $4y$ and $3y$ .

$9x - 4y = 2000 \quad \text{(1)}$ $7x - 3y = 2000 \quad \text{(2)}$

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No Solution Case

Using elimination on parallel lines.

Problem. Use the elimination method to find all possible solutions of the following pair of linear equations:

$2x + 3y = 8 \quad \text{(1)}$ $4x + 6y = 7 \quad \text{(2)}$

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Worksheet

Reversing Digits

Set up and solve a digit problem using elimination.

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Let the ten's and the unit's digits in the first number be $x$ and $y$ , respectively, so the number is $10x + y$ . When the digits are reversed, the number becomes $10y + x$ . According to the problem, the sum of these numbers is 66, which gives the equation $(10x + y) + (10y + x) = 66$ . Simplifying this equation by combining terms to get $11(x + y) = 66$ and dividing by 11 yields the first linear equation: . We are also given that the digits differ by 2, which gives us the second equation $x - y = 2$ (assuming $x > y$ ). By applying the elimination method and adding the two simplified equations together, we eliminate $y$ to get $2x = 8$ , which means $x =$ . Substituting this value back into the first equation, we find that $y =$ , giving us the final two-digit number 42.

← previousSolving By Substitution Method next →Chapter Review: Pair Of Linear Equations

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Eliminating 'y'

Solve a word problem using elimination.

Problem. The ratio of incomes of two persons is $9 : 7$ and the ratio of their expenditures is $4 : 3$ . If each of them manages to save ₹ 2000 per month, find their monthly incomes.

Let the incomes be $9x$ and $7x$ . Let expenditures be $4y$ and $3y$ .

$9x - 4y = 2000 \quad \text{(1)}$ $7x - 3y = 2000 \quad \text{(2)}$

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No Solution Case

Using elimination on parallel lines.

Problem. Use the elimination method to find all possible solutions of the following pair of linear equations:

$2x + 3y = 8 \quad \text{(1)}$ $4x + 6y = 7 \quad \text{(2)}$

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Worksheet

Reversing Digits

Set up and solve a digit problem using elimination.

0 of 3 blanks filled