Solve the system by addition (elimination).EXAMPLESolve the system by addition: Strategy T...

Solve the system by addition (elimination).

EXAMPLE

Solve the system by addition:

Strategy To use the addition method to solve this system, we will multiply both sides of the second equation by 2 and add the equations. This will eliminate the terms involving the variable x.

WHY This will give one equation involving only the variable y.

Solution

Step 1 This is the system discussed in Example. In this example, we will solve it by the addition method. Since both equations are already written in general form, step 1 is unnecessary.

Step 2 We note that the coefficient of x in the first equation is 4. If we multiply both sides of the second equation by 2, the coefficient of x in that equation will be −4. Then the coefficients of x will be opposites.

Step 3 When these equations are added, the terms involving x drop out (or are eliminated), and we get an equation that contains only the variable y.

Step 4 Solve the resulting equation for y.

Step 5 To find x, we substitute −3 for y in either of the original equations and solve for x. If we use the first equation, we have

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Step 6 The solution is (4, −3). The check was completed in Example.

EXAMPLE

Solve the system by substitution:

Strategy We will use the substitution method. Since the system does not contain an equation solved for x or for y, we must choose an equation and solve it for x or y. It is easiest to solve for y in the first equation, because y has a coefficient of 1 in that equation.

WHY Solving 4x + y = 13 for x or solving −2x + 3y = −17 for x or for y would involve working with cumbersome fractions.

Solution

Step 1 We solve the first equation for y, because y has a coefficient of 1.

Step 2 We then substitute −4x + 13 for y in the second equation of the system. This step will eliminate the variable y from that equation. The result will be an equation containing only one variable, x.

Step 3 To find y, we substitute 4 for x in the substitution equation and simplify:

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Step 4 The solution is (4, −3). The graphs of the equations of the given system would intersect at the point (4, −3).

Step 5 To verify that this result satisfies both equations, we substitute 4 for x and −3 for y into the original equations of the system and simplify.

Check:

Since (4, −3) satisfies both equations of the system, it checks.