4. Solve the following system of equations by

a) Graphing

Solve each equation for   y.   For the first one,

2x + y = 1

first, transpose the   x   term to the other side of the equation.

y = 2x + 1

We do the same thing with the other equation.

-2y = -x = 7

but here we also have a coefficient to remove. Divide both sides by -2.

Make up a table of x's and y's for each equation, and graph them both.

and we see that the solution is   x = -1   and  y = 3.

Check

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if we substitute these numbers into the first equation, we get

2(-1)+ (3) = 1

-2 + 3 = 1

which checks.

In the second equation we get

(-1) - 2(3) = -7

-1 - 6 = -7

so they check in both equations

b) Substitution.

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Solve one of the equations for one of the unknowns. In this case, it would be best to solve the second equation for   x.

x = 2y - 7

Substitute this solution into the other equation

2(2y - 7) + y = 1

This gives us an equation with only one unknown. Solve this equation. First, remove parentheses.

4x - 14 + y = 1

Combine the   y   terms on the left and transpose the   -14   to the other side.

5y = 1 + 14

or

5y = 15

Divide both sides by 5.

y = 3

which is what we got the other way.

To find   x,   substitute this solution into the equation where we had solved for   x   as a function of   y.

x = 2y - 7

and get

x = 2(3) - 7

or

x = -1

which is the same solution as we got before.

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Multiply the equations by suitable numbers so that the coefficients of the y terms match up. In this case we could simply multiply the top equation by   2.

or

x = -1

To solve for   y,   substitute this solution into either of the original equations and solve for   y.   The best equation would be the first one

2(-1) + y = 1

or

-2 + y = 1

so all we have to do to solve for y is to add 2 to both sides.

y = 3