Solving Systems of Equations using Substitution

To solve a system of equations without graphing, you can use the substitution method shown in the Example below. In general, if you solve a system of equations and the result is a true statement, such as - 5 = -5, the system has infinitely many solutions; if the result is a false statement, such as - 5 = 7, the system has no solution .

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Example

Use substitution to solve the system of equations x + y = 1 and 2x + y = - 1.

Step 1: Solve one of the equations for x or y.

x + y = 1 Solve the first equation for x since the coefficient of x is 1.

x = 1- y

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Step 2: Substitute this value into the other equation.

Step 3: Solve this equation.

2 - 2y + y = -1 Solve for y.

-y = -3 or y = 3

Step 4: Find the value of the other variable using substitution into either equation.

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The solution to the system is ( - 2, 3).

Check: Substitute -2 for x and 3 for y in each of the original equations and check for true statements.

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