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Method of Elimination by Substitution
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Linear Equation in Two Variables: Elimination by Substitution Method
The method of solving "by substitution" works by solving one of the equations for one of the variables, and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

To solve a system of two linear equations in two variables,
Step 1. Solve one of the equations for one of the variables.
Step 2. Substitute the expression for the variable chosen in step 1 into the other equation.
Step 3. Solve the resulting equation in one variable.
Step 4. Substitute the value obtained in step 3 into the equation obtained in step 1 and solve to obtain the value of the other variable.
Step 5. Check the solution in both equations.
Step 6. Write the solution as an ordered pair.
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Example: Solve the following systems of equations by substitution method
(i) 3x - 5y = -1
x - y = - 1
(ii) x + 2y
2x - 3y = 12
(i) The given system of equations is
3x - 5y = -1 ...(i)
x - y = -1 ...(ii)
From (ii), we get
y = x + 1
Substituting y = x + 1 in (i), we get
3x - 5(x + 1) = -1
=> - 2x - 5 = -1
=> -2x = 4
=> x = -2
Putting x = -2 in y = x + 1, we get y = -1.
Hence, the solution of the given system of equations is x = -2, y = -1.

(ii) The given system of equations is
2x + 3y = 9 ...(i)
3x + 4y = 5 ...(ii)
From equation (i), we get
3y = 9 - 2x
=> y = (9 - 2x)/3
Substituting y = (9 - 2x)/3 in equation (ii), we get
3x + 4((9 - 2x)/3) = 5
=> x + 36 = 15
=> x = -21
Putting x = -21 in y = (9 - 2x)/3, we get
y = (9 + 42)/3 = 17
Hence, the solution of the given system of equation is x = -21, y = 17
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