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There are two main methods for solving simultaneous equations.
• Elimination Method (Addition/Subtraction Method)
To eliminate one of the variables, by adding or subtracting the equations.(If needed, multiply each equations by a constant.)
• Substitution Method
To eliminate one of the variables by substitution.
In this page I will illustrate the Substitution Method for solving simultaneous linear equations. |
2 equations with 2 variables
1 equation with 1 variable4x=12
1 solutionx=3
2 solutionsx=3 y=−1 If you can eliminate one of the two variables and get one equation for one variable, you can solve the equation. 4x=12 → x=3 Thus, the methods for solving simultaneous equations are, in other words, the method for eliminating one variable. |
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The Substitution Method
1.If one of the equations is already solved for y, substitute the equation into the other equation.
(More precisely, substitute the right side of the equation into the other equation.) 
ExampleSolve the following simultaneous equations.
Substitute (1) into (2)
2x+3(2x+1)=11 … One equation with x only. 2x+6x+3=11 … It’s easy to solve. 8x=8 x=1 …(3) Substitute (3) into (1) and solve for y. y=2×1+1=3 …(4) x=1, y=3 …(Answer) |
If one of the equations is already solved for x,the situation is the same with x.
ExampleSolve the following simultaneous equations.
Substitute (1) into (2)
3(2y−1)−2y=5 6y−3−2y=5 4y=8 y=2 …(3) Substitute (3) into (1) and solve for x. x=2×2−1=3 …(4) x=3, y=2 …(Answer)
Notice :
On doing the substitution, you had better to set up the parentheses to avoid the mistakes.
• 2x+3·2x+1=11 ← Incorrect • 2x+3(2x+1)=11 ← Correct |
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2.If one of the coefficients of y equals 1 or −1, you can easily rewrite the equation in ”y=” form.
e.g.
3x+y=2 → y=−3x+2 3x−y=2 → −y=−3x+2 → y=3x−2
ExampleSolve the following simultaneous equations.
Solve (1) for y.
y=−3x+1 …(3) Substitute (3) into (2)
Substitute the new equation to the other equation. Otherwise, it will never work.
5x+3(−3x+1)=−13x+(−3x+1)=1 ← We have nothing to gain. 5x−9x+3=−1 −4x=−4 x=1 …(4) Substitute (4) into (3) and solve for y.
You can also substitute (4) into (1), but it’s easy to substitute (4) into (3).
y=−3×1+1=−2x=1, y=−2 …(Answer) |
The situation is the same with x.
ExampleSolve the following simultaneous equations.
Solve (1) for x.
x=−2y−1 …(3) Substitute (3) into (2) 4(−2y−1)+3y−6=0 −8y−4+3y−6=0 −5y=10 y=−2 …(4) Substitute (4) into (3) and solve for x. x=−2×(−2)−1=3 …(4) x=3, y=−2 …(Answer) |
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3.Even if no variable has a coefficient of 1 or −1, you can rewrite the equation in ”x=” or ”y=” form. But in this case the obtained equation has the coefficients with fractions, and the equations become a little complicated.
e.g.
3y=2x+4 → y=x+ 3x−2y=5 → −2y=−3x+5 → y=x−
ExampleSolve the following simultaneous equations.
Solve (1) for y.
4y=−3x−2 y=−x− …(3) Substitute (3) into (2) 5x+6(−x−)=−4 5x−x−3=−4 10x−9x−6=−8 x=−2 …(4) Substitute (4) into (3) and solve for y. y=−×(−2)−=−=1 x=−2, y=1 …(Answer) |
The simultaneous linear equations can be solved either the Substitution Method or the Addition/Subtraction Method. You can choose whichever method you like.
When no variable has a coefficient of 1 or −1, by using the Addition/Subtraction Method you can avoid fractional coefficients.
By the Addition/Subtraction Method :
(1)×3−(2)×2
9x+12y=−6 …(1)×3
−x=2−) 10x+12y=−8 …(2)×2 x=−2 …(3) Assign (3) to (1) −6+4y=−2 4y=4 y=1 Therefore, x=−2, y=1 |
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QuestionSolve the following simultaneous equations by the substitution method. (Fill in the blanks and click on the Check button. ) |
