Math Pacing Substitution 1.Today Tom has $100 in his savings account and plans to put $25 in the...

Substitution

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?

2. Which is the point of intersection of the lines described by:y = 3x – 1 and y = 5?

Let x = the number of weeksLet y = the total amount savedTom: y = 25x + 100

Maria: y = 50x

Savings Accounts

Number of Weeks

Tot

al S

aved

In four weeks they will both have $200

Substitution

Substitution

The exact solution of a system of equations can be found by using algebraic methods.

One such method is called substitution.

Use substitution to solve the system of equations.

Since substitute 4y for x in the second equation.

Second equation

Combine like terms.

Divide each side by 15.

Simplify.

Solve Using Substitution

Simplify.1

Use to find the value of x.

First equation

Simplify.

Answer: The solution is (20, 5).



Answer: (1, 2)


First equation

Simplify.

Subtract 4x from each side.

Solve the first equation for y since the coefficient of y is 1.


Solve for One Variable, then Substitute

Find the value of x by substituting for y in the second equation.

Second equation

Distributive PropertyCombine like terms.Add 36 to each side.Simplify.

Divide each side by 10.

Simplify.


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 verifiesthe solution.



Answer: (–3, 2)



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



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.

1.4

2

yx

yx 2.

21

41

24

xy

yx 3.834

1.04.03.0

sr

rs

42 yy

4 y

4y

44 x

8x

(8, – 4)

24 xy

2244 xx

2244 xx

22 TRUE

Infinitely Many Solutions

143 rs

8144 rr

8144 rr

81FALSE

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.

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 or 100% gold. Similarly, 18-

karat gold is 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.

Ounces of Gold

4yxTotal Ounces

14-karat gold12-karat gold18-karat gold

The system of equations is and

Use substitution to solve this system.


First equationSubtract y from each side.Simplify.

Distributive Property

Second equation


Combine like terms.

Subtract 3 from each side.

Simplify.

Multiply each side by –4.

Simplify.


First equation

Simplify.

Subtract from each side.

Answer: ounces of the 18-karat gold and ounces

of the 12-karat gold should be used.


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?

Answer: 5mL of 10% solution, 5mL of 40% solution


Let x = the number of milliliters of 10% HCl solutionLet y = the number of milliliters of 40% HCl solution

)10(25.04.01.0

10

yx

yx yx 10

5.24.0101.0 yy

5.24.01.01 yy

5.13.0 y 5y5x

SubstitutionComplementary 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 XLet y = the measure of Y yx

yx

2

90

902 yy

903 y30Ymeasure

60Xmeasure30y

SubstitutionJohn 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 ageLet y = Sally’s age 48

6

yx

yx

486 yy

4862 y422 y21y

Sally is 21 years old.John is 27 years old.

Math Pacing Substitution 1.Today Tom has $100 in his savings account and plans to put $25 in the...

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