4. Solve the following system of equations by

 

2x+3y=3,3x-2y=7

and check your answer

a) Graphing

 

Solve each equation for y. With the first equation

 

2x+3y=3

 

transpose the x term,

 

3y=-2x+3

 

and divide by the coefficient of y.

y=-(2/3)x+1

With the other equation,

 

3x-2y=7

 

transpose the x term,

 

-2y=-3x+7

 

and divide by the coefficient of the y.

 

y=(3/2)x-7/2

 

Now graph both equations in the same graph.

 

Graphs of both equations

 

We can see that the x coordinate is somewhere around 2, and

the y coordinate is between 0 and -1. But we will need to use other,

computational methods to be more precise.

 

b) Substitution

top

2x+3y=3,3x-2y=7

 

Solve one of the equations for one of the unknowns. It would

be best to solve the first equation for y for two reasons. Most

importantly, if we solve the first equation for y, we will have fewer

fractions to work with. The other reason is that we have already

solved the first equation for y in the process of graphing.

 

y=-(2/3)x+1

 

Now substitute this solution in for y in the other equation.

 

3x-2(-(2/3)x+1)=7

 

This gives us an equation with only one unknown. We can solve

this equation. Remove parentheses.

 

3x+(4/3)x-2=7

 

Clear denominators. Multiply both sides by 3.

 

3(3x+(4/3)x-2)=3(7)

 

or

 

9x+4x-6=21

 

Combine the x terms and transpose the 6.

 

13x=21+6

 

or

 

13x=27

 

Divide both sides by 13.

x=27/13

 

or

 

x=2+1/13

which, as we suspected, is very close to 2.

 

To find y, the simplest way would be to substitute this solution

into the equation where y is expressed as a function of x.

 

 

y=-(2/3)(27/13)+1

 

After simplifying, find common denominators.

 

y=-18/13+13/13

or

 

y=-5/13

which is also about what we expected.

 

Check

top

If we substitute these values in the first equation,

 

2x+3y=3

 

we get

 

2(27/13)+3(-5/13)=3

or

 

54/13-15/13=3

 

39/13=3

which checks.

 

The other equation

 

3x-2y=7

 

becomes

 

3(27/13)-2(-5/13)=7

 

 

81/13+10/13=7

 

91/13=7

and they both check.

 

c)

top

Addition

2x+3y=3,3x-2y=7

 

In order to get the coefficients of the y terms to match up, we

could multiply the top equation by 2 and the other equation by -3.

 

2x+3y=3M2:4x+6y=6,3x-2y=7M3:9x-6y=21.Add:13x=27

 

and we get

 

x=27/13

 

To find y, we substitute this solution into either of the original

equations. If we substitute this into the first equation, we get

 

2(27/13)+3y=3

 

and solve for y.

 

Simplify.

 

 

54/13+3y=3

 

transpose the 54/13 to the other side of the equation and find

common denominators.

 

3y=39/13-54/13

or

 

3y=-15/13

Divide both sides by 3.

 

(3y)/3=(-15/13)/3

 

and we get the same answer as before.

 

y=-5/13

 

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