4. Solve the following system of equations by

 

 

and check your answer

 

2x+y=1,x-2y=-7

 

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.

 

y=x/2+7/2

 

 

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

both.

 

Graphs of both equations

 

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

 

Check

top

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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2x+y=1,x-2y=-7

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

 

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.

 

c) Addition

top

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.

 

2x+y=1M(2):4x+2y=2,x-2y=-7;5x=-5

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

 

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