3. Solve the following system of equations by

and check your answer

4x+3y=3,3x+2y=7

a) Graphing.

Solve the first equation for y

 

4x+3y=3

 

Transpose the 4x to the other side of the equation.

 

3y=-4x+3

 

Then divide both sides by 3.

 

 

y=-(4/3)x+1

 

Do the same thing for the other equation.

 

3x+2y=7

 

Transpose

 

2y=-3x+7

 

Divide

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

 

Now graph both equations in the same picture.

 

Computations and graphs of the two equations

 

 

 

The lines look very close to being parallel, but they aren't. The

slope of the first line is -4/3, and the slope of the second line is -3/2.

Since these two numbers are fairly close together, these lines look

like they are almost parallel, but since the slopes are not exactly

equal, the lines are not exactly parallel. The second line has a slightly

steeper slope than the first line, so they will meet if we extend the

graphs far enough to the right. For the first line, a slope of -4/3

means that you go down 4 units (up -4 units) when you go over 3

units. For the second line, a slope of -3/2 menas that you go down 3

units when you go over 2 units.

Graphs of the two equations extending to where the lines meet

 

 

and the lines finally meet at the point (15, -19).

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Check: Let x = 15 and y = -19

 

4(15)+3(-19)=3

 

60-57=3

 

And in the second equation

 

3(15)+2(-19)=7

 

45-38=7

 

So both answers check.

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In this case while we are quite fortunate that the lines meet at

a point that has whole number coordinates, we have to go for quite a

way before the lines actually meet. In this case one of the

computational methods might work better

 

b) Substitution

 

4x+3y=3,3x+2y=7

 

Solve one of the equations for one of the unknowns. In this case it

will be simplest to solve the first equation for y; first, because that is

the unknown for which we will have the fewest fractions when we

solve, and second, because we have already solved the first equation

for y when we graphed the two equations.

 

y=-(4/3)x+1

 

Substitute this in for y in the other equation

 

3x+2(-(4/3)x+1)=7

 

Remove parentheses.

 

3x-(8/3)x+2=7

 

Clear fractions by multiplying both sides by 3.

 

3(3x-(8/3)y+2)=3(7)

 

9x-8x+6=21

 

x+6-6=21-6, x=15

 

The simplest; way to find y at this point is to substitute this

solution into the equation where we solved for y as a function of x.

 

y=-(4/3)(15)+1

 

Fortunately, this simplifies

 

 

y=-20+1

 

 

y=-19

 

 

 

 

and we get the same solution which we have previously shown to

check.

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c) Addition

 

Since the coefficients of neither of the variables match up, we

need to multiply the equations by suitable numbers. In this case,

multiply the first equation by 3 and the second equation by -4.

 

(12x+9y=9)-(12x+8y=28)=(y=-19)

 

 

We get our solution for y very quickly with this method. If one

is using only this method, then one will not have either equation

solved for either unknown at this point, so we will have to substitute

this solution into one of our original equations. If we substitute it

into the first equation we get

 

 

4x+3(-19)=3

 

4x-57+57=3+57,4x=60

 

x=15

 

The same solution we got before.

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