Substitution
Let x = the number of weeks
Let y = the total amount saved
Tom: y = 25x + 100
Maria: y = 50x
In four weeks they will both have $200
Total Saved
1. Today Tom has $100 in his savings account and plans to put $25
in the account every week. Maria has nothing in her account but
plans to put $50 in her account every week. In how many weeks
will they have the same amount in their accounts? How much
will each person have saved at that time? Savings Accounts
Number of Weeks
2. Which is the point of intersection of the lines described by:
y = 3x – 1 and y = 5?
Substitution
Substitution
The exact solution of a system of equations can be
found by using algebraic methods.
One such method is called substitution.
Solve Using Substitution
Use substitution to solve the system of equations.
substitute 4y for x in the second equation.
Since
Second equation
1
Simplify.
Combine like terms.
Divide each side by 15.
Simplify.
Use
Solve Using Substitution
to find the value of x.
First equation
Simplify.
Answer: The solution is (20, 5).
Solve Using Substitution
Use substitution to solve the system of equations.
Answer: (1, 2)
Solve for One Variable, then Substitute
Use substitution to solve the system of equations.
Solve the first equation for y since the coefficient of y is 1.
First equation
Subtract 4x from each side.
Simplify.
Solve for One Variable, then Substitute
Find the value of x by substituting
for y in the
second equation.
Second equation
Distributive Property
Combine like terms.
Add 36 to each side.
Simplify.
Divide each side by 10.
Simplify.
Solve for One Variable, then Substitute
Substitute 5 for x in either equation to find the value of y.
First equation
Simplify.
Subtract 20
from each side.
Answer: The solution is (5, –8).
The graph verifies
the solution.
Solve for One Variable, then Substitute
Use substitution to solve the system of equations.
Answer: (–3, 2)
Inconsistent or Dependent Equations
Use substitution to solve the system of equations.
Solve the second equation for y.
Second equation
Subtract x from each side.
Simplify.
Substitute
for y in the first equation.
First equation
Distributive Property
Simplify.
Inconsistent or Dependent Equations
The statement
is false. This means there are no
solutions of the system of equations. This is true because
the slope-intercept form of both equations show that the
equations have the same slope, but different y-intercepts.
That is, the graphs of the lines are parallel.
Answer: no solution
Inconsistent or Dependent Equations
Use substitution to solve the system of equations.
Answer: infinitely many solutions
Substitution
Solution Possibilities for Systems of Equations
1. The variables have exactly one value and the system
has exactly one solution.
2. The solution results in a true statement and the
system has infinite solutions.
3. The solution results in a false statement and the
system has no solution.
Substitution
Use substitution to solve each system of equations. If the
system does not have exactly one solution, state whether it
has no solution of infinitely many solutions.
1. x   2 y
x y 4
2. 4 x  y  2
1
4
y  x
3. 0 . 3 s   0 . 4 r  0 . 1
1
2
4r  3s  8
 2y  y  4
y  4x  2
3s  4r  1
 y 4
4 x  4 x  2   2
4 r   4 r  1  8
y  4
4x  4x  2  2
4r  4r  1  8
x   4   4
22
18
x 8
TRUE
(8, – 4) Infinitely Many Solutions
FALSE
No Solution
Substitution
Sometimes it is helpful to organize the data
before solving a problem.
Some ways to organize data are to use tables,
charts, different types of graphs or diagrams.
Write and Solve a System of Equations
Gold Gold is alloyed with
different metals to make it hard
enough to be used in jewelry. The amount of gold
present in a gold alloy is measured in 24ths called
karats. 24-karat gold is
karat gold is
or 100% gold. Similarly, 18-
or 75% gold. How many ounces of 18-
karat gold should be added to an amount of 12-karat
gold to make 4 ounces of 14-karat gold?
Write and Solve a System of Equations
Let
the number of ounces of 18-karat gold and
the number of ounces of 12-karat gold. Use the table
to organize the information.
18-karat gold 12-karat gold 14-karat gold
Total Ounces
x
y
4
Ounces of Gold
The system of equations is
and
Use substitution to solve this system.
Write and Solve a System of Equations
First equation
Subtract y from each side.
Simplify.
Second equation
Distributive Property
Write and Solve a System of Equations
Combine like terms.
Subtract 3 from each side.
Simplify.
Multiply each side by –4.
Simplify.
Write and Solve a System of Equations
First equation
Subtract
from each side.
Simplify.
Answer:
ounces of the 18-karat gold and
of the 12-karat gold should be used.
ounces
Write and Solve a System of Equations
Chemistry Mikhail needs a 10 milliliters of 25% HCl
(hydrochloric acid) solution for a chemistry
experiment. There is a bottle of 10% HCl solution and
a bottle of 40% HCl solution in the lab. How much of
each solution should he use to obtain the required
amount of 25% HCl solution?
Let x = the number of milliliters of 10% HCl solution
Let y = the number of milliliters of 40% HCl solution
x  y  10
x  10  y
0 . 1 x  0 . 4 y  0 . 25 (10 )
1  0 .1 y  0 .4 y  2 .5
0 . 110  y   0 . 4 y  2 . 5
0 .3 y  1 .5
y5
x5
Answer: 5mL of 10% solution, 5mL of 40% solution
Substitution
Complementary angles are two angles whose measures have the
sum of 90°. Angles X and Y are complementary and the measure of
angle X is two times bigger than the measure of angle Y. Find the
measures of angles X and Y.
Let x = the measure of X
Let y = the measure of Y
2 y  y  90
3 y  90
y  30
x  y  90
x  2y
measure  Y  30
measure  X  60


Substitution
John is 6 years older than Sally. Together, their ages add up to 48.
How old is John? How old is Sally?
Let x = John’s age
Let y = Sally’s age
y  6  y  48
2 y  6  48
2 y  42
y  21
x  y6
x  y  48
Sally is 21 years old.
John is 27 years old.