3. Solve the following system of equations by

and check your answer

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.

Do the same thing for the other equation.

3x + 2y = 7


2y = -3x + 7


Now graph both equations in the same picture.

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. Notice that the lines are getting closer together as they go down and to the right, so they will meet if we extend the graphs far enough in that direction.

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   means that you go down 3 units when you go over 2 units.

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


Check: Let x = 15 and y = -19

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

60 - 57 = 3

and it checks. In the second equation

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

45 - 38 = 7

So both answers check.


While, in this case, 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. Tghis is another example where one of the computational methods might work better.

b) Substitution

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.

Substitute this in for y in the other equation

Remove parentheses.

Clear fractions by multiplying both sides by 3.

9x - 8x + 6 = 21

x + 6 = 21

Subtract 6 from both sides.

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.

Fortunately, this simplifies

y = -20 + 1

y = -19

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


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.

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 = 3

Add 57 to both sides.

4x = 60

Divide both sides by 4.

x = 15

The same solution we got before, which we have seen checks.


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