Algebra:

A man lived one-fourth of his life as a boy in Baltimore, one-fifth of his life as a young man in San Francisco, one-third of his life as a man in Manitoba, and the last thirteen years of his life in Thurmont. How old was the man when he died? Write down your algebraic equation, your work, and box your answer.

Read the entire problem through. Note that not all information given is relevant.

1. Write Given, Assign Variables, Sketch and Label Diagram

1. Whenever you write a variable, you must write what that variable means.

2. What are the quantities? Assign variable(s) to quantities.

3. If possible, write all quantities in terms of the same variable.

2. Write Formulas / Equations

What are the relationships between quantities?

3. Substitute and Solve

Communication: All of your work should communicate your thought process (logic/reasoning).

4. Check Answer, then Box Answer

Word ProblemsWord Problems

X = X/4 + X/5 + X/3 + 13

60X = 15X + 12X + 20X + 78060X = 47X + 78013X = 780

X = 60 years

ALGEBRAIC SOLUTION:Define the unknowns: X = man’s total age. X/4 = years as a boy X/5 = years as a youth X/3 = years as a manWrite an equation:

Solve the equation:

Systems of EquationsSystems of Equations

• Because two equations impose two conditions on the variables at the same time, they are called a system of simultaneous equations.

• When you are solving a system of equations, you are looking for the values that are solutions for all of the system’s equations.

• Methods of Solving:1. Graphing2. Algebra:

1. Substitution2. Elimination

1. Addition-or-Subtraction2. Multiplication in the Addition-or-Subtraction Method

• Solve the following system by graphing:

y = x2

y = 8 – x2

What is the solution?

(2, 4) and (-2,4)

Systems of EquationsSystems of Equations

• Solve the following system algebraically:1) y = x2

2) y = 8 – x2

Substitute equation 1 into equation 2 and solve: x2 = 8 – x2

2x2 = 8 x2 = 4 x = 2 and -2

Now substitute x-values into equation 1 to get y-values:when x = 2, y = 4when x = -2, y = 4

Solution: (2, 4) and (-2,4)

Systems of EquationsSystems of Equations

• Systems of equations can have: One Solution

Multiple Solutions

No Solutions

Systems of EquationsSystems of Equations

• Solve using the same method as single equation problems:

1. Write Given, Assign Variables, Sketch and Label Diagram1. Whenever you write a variable, you must write what that

variable means.2. What are the quantities? Assign variable(s) to quantities.3. If possible, write all quantities in terms of the same

variable.2. Write Formulas / Equations

What are the relationships between quantities?3. Substitute and Solve

Communication: All of your work should communicate your thought process (logic/reasoning).

4. Check Answer, then Box Answer

Systems of Equations – Word ProblemsSystems of Equations – Word Problems

Examples:

Systems of Equations – Word ProblemsSystems of Equations – Word Problems

Algebra A:

Jenny and Kenny together have 37 marbles, and Kenny has 15. How many does Jenny have? (Solve algebraically, then graphically to check.)Algebra B:

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? Algebra C:

Three times the width of a certain rectangle exceeds twice its length by three inches, and four times its length is twelve more than its perimeter. Find the dimensions of the rectangle.

Classwork:

Systems of Equations – Word ProblemsSystems of Equations – Word Problems

Algebra A:

The perimeter of a rectangle is 54 centimeters. Two times the altitude is 3 centimeters more than the base. What is the area of the rectangle?

Algebra B:

The sum of the digits in a two-digit numeral is 10. The number represented when the digits are reversed is 16 times the original tens digit. Find the original two-digit number.

Hint: Let t = the tens digit in the original numeral and u = the units digit in the original numeral.

• Systems of Equations:– Use the multiplication / addition-or-subtraction

method to simplify and/or solve systems of equations:• Eliminate one variable by adding or

subtracting corresponding members of the given equations (use multiplication if necessary to obtain coefficients of equal absolute values.)

NotesNotes

• Area of a circle: pi*r2

• Volume of a sphere: (4/3)*pi*r2

• Volume of a cylinder: h*pi*r2

• Surface area of a sphere: 4*pi*r2

• Surface area of a cylinder: 2*pi*r2 + 2*pi*r• Surface area of a rectangular prism: 2*a*b + 2*a*c + 2*b*c • Area of a triangle: (1/2)*b*h• Volume of a pyramid: (1/3)*Abase*h

• Area of a circle:• Volume of a sphere:• Volume of a cylinder:• Surface area of a sphere:• Surface area of a cylinder:• Surface area of a rectangular prism:• Area of a triangle:• Volume of a pyramid:

Geometry ReviewGeometry Review