Algebra Textbook Volume 1 Student Edition wc

The object of this method is to _add_ the two equations together so that one of the two
variables is eliminated. It makes no difference if x drops out or y drops out, as long as an
equation with only _one_ variable is left.
Sometimes this will happen with no preliminary step and sometimes a preliminary step is
necessary. After the equations are added and one equation with one variable is left, it is solved.
The value for that variable is then substituted back into either original equation and the value of
the remaining variable can be determined. The resulting ordered pair is the _solution__ to the
system of equations. This ordered pair is the one pair that works in both original equations. If the
equations were graphed, the ordered pair would be the point where the lines _intersected__.

7.4 A Cramer’s Rule
Earlier in this chapter, three methods of solving systems of two equations with two unknowns
were explained and practiced: graphing, substitution and elimination. There is another method
called Cramer’s Rule which is simple, very straightforward and easy to remember. It requires
basic arithmetic, but no algebraic manipulation. Cramer’s Rule is an especially efficient method
to use whenever _fractions__ are involved.
As the name suggests (Cramer’s Rule), this method is rule-oriented. The equations will be
solved by following a _step_ by _ step _ procedure. Assume the following two equations are
to be solved using Cramer’s Rule. Note that the equations are written in __standard__
form: Ax + By = C

7.5 A Standard Equation Form
The graph of an equation that can be written in the form Ax+By = C is a _straight_ line and,
therefore, it is referred to as a linear equation.
Solve this equation for y. This means to rearrange it to the form y = ___mx+b__.
This equation clearly shows the relationship between a given x-value and its corresponding y-
value. The set of all ordered pairs fitting this equation will form a straight line. For every ordered
pair on this line the y-value is three more than −2 times the x-value. The line can be labelled y =
−2x+3.

78 | P a g e

In the case of every point above that line, the y-coordinate is _greater__ than three more than
−2 times the x-coordinate. This can be represented as y > −2x+3.

7.6 A Solving System of Linear Inequalities
A system of linear inequalities is a set of several, usually two, linear inequalities. The solution to
such a system is the group or set of all ordered pairs, or corresponding points, that satisfy all of
the linear inequalities. These systems are solved by graphing the system of linear inequalities and
identifying the set of points that are part of both solutions. The final solution is the set of points
where the individual graphs overlap. To identify them on the graph, the shading is usually
_darkened_. Notice that in solving a system of linear equations, the equations will have, at
most, only __one_ point in common. When solving a system of linear inequalities, there is the
possibility of an __infinite_ number of solutions.

7.7 A Linear Programming
Linear programming is a method of solving real-life problems using linear _inequalities_.
This method will be used here to help a business decide on the best manufacturing plan to give
the greatest _profit_. In the simple application of linear programming considered here, a
company can produce two similar items. These two items require different amounts of labor and
materials; and the company has limits on the amount of each available. The inequalities
describing the limitations due to labor and materials are called the _constraints_. The
solution to the system of linear inequalities, the constraints, make up what is called the feasibility
region. All points in this feasibility region meet the limitations placed on the manufacturer of the
two items. In these examples, the feasibility region will always have four corner points.
Suppose a company would like to identify the manufacturing plan to give the greatest profit. The
maximum value for the profit expression will occur at one of the __four_ corner points of the
feasibility region. The coordinates of those corner points are identified and then used in the profit
expression. The coordinates giving the greatest profit, then, indicate the best manufacturing plan-
the number of each item to be produced each week.

79 | P a g e

Chapter 7 Review A

C7 Review A Name ____________________________

I. Solving two equations with two unknowns

Solve each of the following systems of two equations with two unknowns using one of the
methods explained in this chapter. The graph is there if needed.

1) 3 + 2 = −9 y
− 2 = −11

x

2) = 1 + 3 Method used: __________________ Solution: ________________
2 y
= 3 − 2
x

Method used: __________________ Solutiony: ________________

80 | P a g e

C7 Review A (cont.) Name ________________y____________
3) 2 + 4 = −1 x
3 + 2 = 1

4) = 2 − 12 Method used: __________________ Solution: ________________
4 + 3 = 7 y

x

5) 5 + 3 = 3 Method used: __________________ Solution: ________________
5 + 2 = 7 y

x

Method used: __________________ Solution: ________________
81 | P a g e

C7 Review AI (cont.) Name ____________________________
6) 3 − 4 = −3 y
+ 4 = 1
x

7) = −31 32 + − 1 Method used: __________________ Solution: ________________
= 2 y

x

Method used: __________________ Solution: ________________
y

8) = 2 + 18
3 + 2 = −13

x

Method used: __________________ Solution: ________________
82 | P a g e

C7 Review A (cont.) Name ____________________________

II. Solving Linear Inequalities

Graph the solutions to the following problems: y
9) 3 + 4 ≤ 12

x

y

10) 2 + > −1
3 − > −4
x

83 | P a g e