1. Algebra 7-2 Substitution
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?
Which is the point of intersection of the lines described by: y = 3x – 1 and y = 5?
Savings Accounts
Let x = the number of weeks
Let y = the total amount saved
Total Saved
Tom: y = 25x + 100
Maria: y = 50x
In four weeks they will both have $200
Number of Weeks
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2. Algebra 7-2 Substitution
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3. Algebra 7-2 Substitution
The exact solution of a system of equations can be found by using algebraic methods.
One such method is called substitution.
Substitution
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4. Solve Using Substitution
Use substitution to solve the system of equations.
Since substitute 4y for x in the second equation.
Second equation
Simplify. 1
Combine like terms.
Divide each side by 15.
Simplify.
Example 2-1a

5. Solve Using Substitution
Use to find the value of x.
First equation
Simplify.
Answer: The solution is (20, 5).
Example 2-1a

6. Solve Using Substitution
Use substitution to solve the system of equations.
Answer: (1, 2)
Example 2-1b

7. 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.
Example 2-2a

8. 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.
Example 2-2a

9. Solve for One Variable, then Substitute
Substitute 5 for x in either equation to find the value of y.
Answer: The solution is (5, –8). The graph verifies the solution.
First equation
Simplify.
Subtract 20 from each side.
Example 2-2a

10. Solve for One Variable, then Substitute
Use substitution to solve the system of equations.
Answer: (–3, 2)
Example 2-2b

11. 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.
Example 2-3a

12. 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
Example 2-3a

13. Inconsistent or Dependent Equations
Use substitution to solve the system of equations.
Answer: infinitely many solutions
Example 2-3b

14. Algebra 7-2 Substitution
Solution Possibilities for Systems of Equations
The variables have exactly one value and the system has exactly one solution.
The solution results in a true statement and the system has infinite solutions.
The solution results in a false statement and the system has no solution.
Substitution
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15. Algebra 7-2 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. 2. 3.
(8, – 4)
Infinitely Many Solutions
No Solution
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16. Algebra 7-2 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.
Substitution
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17. 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 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?
Example 2-4a

18. 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 4 y x
Total Ounces
14-karat gold
12-karat gold
18-karat gold
The system of equations is and
Use substitution to solve this system.
Example 2-4a

19. Write and Solve a System of Equations
First equation
Subtract y from each side.
Simplify.
Second equation
Distributive Property
Example 2-4a

20. Write and Solve a System of Equations
Combine like terms.
Subtract 3 from each side.
Simplify.
Multiply each side by –4.
Simplify.
Example 2-4a

21. Write and Solve a System of Equations
First equation
Subtract from each side.
Simplify.
Answer: ounces of the 18-karat gold and ounces of the 12-karat gold should be used.
Example 2-4a

22. 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
Answer: 5mL of 10% solution, 5mL of 40% solution
Example 2-4b

23. Homework Page 380 #30 – 35 (6 problems – 16 points)
Algebra 7-2 Substitution
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
Homework Page 380 #30 – 35 (6 problems – 16 points)
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24. Homework Page 380 #30 – 35 (6 problems – 16 points)
Substitution
Algebra 7-2 Substitution
John is 6 years older than Sally. Together, their ages add up to How old is John? How old is Sally?
Let x = John’s age
Let y = Sally’s age
Sally is 21 years old.
John is 27 years old.
Homework Page 380 #30 – 35 (6 problems – 16 points)
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