**4.3. A Short Quiz to Test Yourself**

Now try a few for yourselves. After you solve the problem , click on the green
link

of your choice to get immediate feedback to your choice.

**Quiz **Solve each of the fol lowing systems using any method you choose .

1.

(4,−1)

(−9, 3)

(−8, 5)

(2, 2)

No solution

{(x, y) | x + 3y = 0}

2.

(4,−1)

(−9, 3)

(−8, 5)

(2, 2)

No Solution

{(x, y) | x − 5y = −3}

3.

(4,−1)

(−9, 3)

(−8, 5)

(2, 2)

No solution

{(x, y) | 2x − y = 9}

4.

(4,−1)

(−9, 3)

(−8, 5)

(2, 2)

No solution

{(x, y) | −x + 3y = 4}

5.

(4,−1)

(−9, 3)

(−8, 5)

(2, 2)

No solution

{(x, y) | 2x+3y = −1}

How did you do?

** Solutions to Examples **

**Example 2.1(a): **We substitute the point (−5, 3) into the equation to get

You can see that the first equation is satisfied, but the second equation is
not.

**Conclusion: **The point (−5, 3) does not satisfy the system.

**Example 2.1(b):** We substitute the point (−3, 4) into the equation to get

You can see that the both equations are satisfied.

**Conclusion: **The point (−3, 4) does satisfy the system.

**Example 4.1(a):** We solve for y in the first equation of
to get

We now substitute into the second equation:

second equation

substitution

combine

and solve

Now substituting x = 6 into the
we get y = −2.

**Solution: **These are consistent equations with a unique solution of
**(6,−2) **, or

the solution set is **{ (6,−2) } .**

**Example 4.1(b):** We solve for y in the first equation of
to

get y = 1− 3x, and substitute into the second equation:

−6x − 2y = −4
second equation

−6x − 2(1 − 3x) = −4 substitution

−6x − 2 + 6x = −5 combine

−2 = −5 simplify (2)

We see that in equation (2) we get a “false” equation. This means that there is

no solution to the system.

**Solution: **This is an inconsistent system of equations, there is no solution to
this

system. The solution set is .

**Example 4.1(c): **We solve for x in the first equation in
to get

x = 6− 3y. Substitute into the second equation:

second equation

substitution

factor out 3

combine

deal with the minus sign , correctly

simplify again (3)

Equation (3) is typical of a dependent system of equations. The equation is
always

true regardless of the value of x . Dependent systems means essentially that
there

are two equations , but each equation is describing the same line .

**Solution: **This is an consistent system of dependent equations. Any point that

satisfies one of the equations automatically satisfies the the other equation.
Therefore,

the solution set is the set of all points on the satisfy either of the two
equations.

We can write **{ (x, y) | x + 3y = 6} **

**Example 4.4(a): **To solve

we simply add the two equations together to get 3y = 9, hence, y = 3. From the

first equation, we see that x = −5 − x = −5 −3 = −8.

**Conclusion: **This is a consistent system of equations with a unique solution of

**(−8, 3) .**

**Example 4.4(b): **As discussed earlier, we multiply the first equation by −3:

given

multiply by −3

add the two equations together

solve for y

From the first equation x = −5 − 2y = −5 − 2(−8) = −5 + 16 = 11

**Conclusions: **This is a consistent system of equations with a unique solution of

**(11,−8) .**