Have a Very
| Substitution
Method for solving linear systems |
- Solve one
of the equations for one of its variables
- Substitute
the expression from Step 1 into the other
equation and solve for the other variable.
- Substitute
the value from Step 2 into the revised equation
from Step 1 and solve.
|
| Linear
Combination Method for solving linear systems |
- Multiply one or
both of the equations by a constant to obtain
coefficients that differ only in sign for one of
the variables.
- Add the revised
equations from Step 1. Combining like terms will
eliminate one of the variables. Solve for the
remaining variable.
- Substitute the
value obtained in Step 2 into either of the
original equations and solve for the other
variable.
|
| Graphing
a System of Linear Inequalities |
To graph a system of linear
inequalities, do the following for each inequality in the
system:
- Graph
the line that corresponds to the inequality. Use
a dashed line for an inequality with < or >
and a solid line for an inequality with <(less
than or equal to) or >(greater
than or equal to).
- Lightly
shade the half-plane that is the graph of the
inequality. Colored pencils may help you
distinguish the different half-panes.
The
graph of the system is the region common to all of the
half-planes. If you used colored pencils, it is the
region that has been shaded with every color.
|
Methods:
|
Solutions:
|
Substitution Method
|
Solve this linear system using
the substitution method: 3x+4y=-4 equation
1
x+2y=-4
equation 2
- Solve Equation 2 for x.
x+2y=2
write equation 2
x=-2y+2
revised equation 2
- Substitute the expression
for x into Equation 1 and solve for y.
3x+4y=-4
write equation 1
3(-2y+2)+4y=-4
substitute -2y +2 for x
y=5
solve for y
- Substitute the value of y
into revised Equation 2 and solve for x.
x=-2y+2
write revised equation 2
x=-2(5)+2
substitute 5 for y
x=-8
simplify
The
solution is (-8,5)
|
Linear Combination
Method
|
Solve the linear system using the
linear combination method: 2x-4y=13 equation 1
4x-5y=8
equation
2
- multiply
the first equation by -2 so that the x
coefficients differ only in sign.
-4x+8y=-26
4x-5y=8
- Add
the revised equations and solve for y.
3y=-18
y=-6
- Substitute
the value of y into one of the original
equations. Solve for x.
2x-4y=13
write
equation 1
2x-4(-6)=13
substitute
-6 for y
2x+24=13
simplify
x=-11/2
solve
for x
The
solution is (-11/2,-6).
|
Graphing a System of
Linear Inequalities
|
Graph the system: y>-3x-1
Inequality
1
y<x+2
Inequality
2
Solution:
Begin
by graphing each linear inequality. Use a different color
for each half-plane. For instance, you can use red for
Inequality 1 and blue for Inequality 2. The graph of the
system is the region that is shaded purple
- shade
the half-plane on and to the right of y=-3x-1
red.
- shade
the half-plane below y=x+2 blue.
- The
graph of the system is the intersection of the
red and blue regions.
|
Word Problem: Functions of Two
Variables
You are sewing mittens to give as
Christmas presents to your family. There are two colors of yarn
that you are using: red and green. The red yarn costs $2 per
bundle and the green yarn costs $1.50 per bundle. To make the
mittens, you buy a sewing machine that costs $25.
Write a model for the
total amount you will spend as a function of the number
of bundles of red and green yarn.
Evaluate the model for
several different amounts of red and green yarn, and
organize your results in a table.
Solution:
Your total cost involves
two variable costs (for the two colors of yarn) and one
fixed cost (for the sewing machine).
total cost= C
(dollars)
red yarn cost= 2
(dollars per pound)
red yarn amount= x
(bundles)
green yarn cost= 1.5
(dollars per pound)
green yarn amount= y
(pounds)
sewing machine cost= 35
dollars
C= 2x + 1.5y + 35
To evaluate the function
of two variables, substitute values of x and y into the
function. For instance, when x=10 and y=20, the total
cost is:
C =2x+1.5y+35 write original
function
=2(10)+1.5(20)+35 substitute for x
and y
=85 simplify
The table shows the total cost
for several different values of x (red yarn) and y (green yarn).
| 0 |
10 |
20 |
30 |
40 |
| 10 |
$70 |
$85 |
$100 |
$115 |
| 20 |
$90 |
$105 |
$120 |
$135 |
| 30 |
$110 |
$125 |
$140 |
$155 |
| 40 |
$130 |
$145 |
$160 |
$175 |
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