4. Solve the following system of equations by

 

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.

With the other equation,

3x - 2y = 7

transpose the   x   term,

-2y = -3x + 7

and divide by the coefficient of the   y.

Now graph both equations in the same graph.

We can see that the x coordinate is somewhere around 2, but when x = 2, the y-coordinate of the first graph is   -1/3,   but the y-coordinate of the second graph is at   -1/2  -   not the same. So the y-coordinate is between   -1/3   and   -1/2   when   x = 2. It's hard to tell the difference between 2/5 and 3/8 by looking at the graph. Which one is it?  We will need to use other, computational methods to be more precise.

b) Substitution

top

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.

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

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

Clear denominators. Multiply both sides by 3.

or

9x + 4x - 6 = 21

Combine the   x   terms and transpose the   6.

13x = 21 + 6

or

13x = 27

Divide both sides by 13.

or

which, as we suspected, is rather 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.

After simplifying, find common denominators.

which is also about what we expected. You will find that 5/13 is between 2/5 and 3/8.

Check

top

If we substitute these values in the first equation,

2x + 3y = 3

we get

or

which checks.

The other equation

3x - 2y = 7

becomes

and they both check.

c) Addition

top

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.

and we get

To find   y,   we substitute this solution into either of the original equations. If we substitute this into the first equation, we get

and solve for   y.

Simplify.

transpose the   54/13   to the other side of the equation, and find common denominators.

Divide both sides by 3.

and we get the same answer as before.

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