Algebra-2 Section 1-3 And Section 1-4

Quiz Simplify 1. Simplify -4y – x + 10x + y 2. Is x = -2 a solution to following equation? 3. Solve.

1-3: Solve Linear Equations

Vocabulary Linear Equation in one variable. 4x – 2 = 64x – 2 = 64x – 2 = 64x – 2 = 6

Definitions: Solution to an equation: the value of the variable that makes the equation “true”. Equivalent equation: has the same solution as the original equation: The solution to both equations is x = 2. They are equivalent equations.

Equality Properties Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Only apply to equations!!! “+, -, x, ÷” by the same number on both sides of the equal sign and you are guaranteed that the next equation is an equivalent equation.

Vocabulary Coefficient The number that multiplies a variable or is “in front of” the variable

2x = 24 Property of Equality Equivalent equation to the one above. 1x = 12 ÷ Property of Equality and Turn coefficients into ones and addends into zeroes so that they disappear! Equivalent equation to the both equations above. inverse prop. Of multiplication

1. 2 = 3 + x Your turn: solve the equation, justify each step

Your turn: solve the equation. Justify each step = 3x + x

Your turn: solve the equation. Justify each step = 2x – 3 + 2x

Your turn: = = -2

Variable on Both Sides 24 - x = 3x + x ÷ 4 6 = x Eliminate variable from one side. 24 = 4x The solution looks like: x = number

x = 3 + 3x Your turn: Solve the following equations 8.

Solving Equations using the Distributive Property 3(5x – 8) = -2(-x +7) – 12x 15x -24 = 2x x Eliminate parentheses using distributive property. Combine “like terms”. 15x -24 = -10x - 14

Solving Equations using the Distributive Property 15x -24 = -10x - 14 Solve using properties of equality. +10x = x 25x -24 = x ÷25÷25

Another example + 4x +6 7x ÷ 7 = 10 7x – 6 = x = 10 ÷ 7

Your Turn: Solve using the Distributive Property

12. Your turn: Solve the following equations (Get rid of parentheses 1 st using the distributive property.)

Solving a multi-variable equation There are an infinite number of combinations. x = 0 x = 0 y = 2 y = 2 z = 2 z = 2 x = 2 x = 2 y = 1 y = 1 z = 2 z = 2 x = 4 x = 4 y = 0 y = 0 z = 0 z = 0

Solving a multi-variable equation x = ? x = ? y = 1 y = 1 z = 2 z = 2 Could you find the value of ‘x’ if I gave you the values of ‘y’ and ‘z’ ? of ‘y’ and ‘z’ ?

Your Turn: Solve for x 15. y = 2, z = 1, x = ? 16. x = 1, y = 3, z = ?

Review What are the +/-/x/÷ properties of equality ? What do the +/-/x/÷ properties of equality guarrantee ? “add 2 to equation on both sides of the equal sign”, etc. The resulting equation will be equivalent to the original equation (it will have the same solution). equation (it will have the same solution).

Section 1-4 Rewrite Formulas and Equations. Section 1-4 Rewrite Formulas and Equations.

Vocabulary Solve for a variable (more then one variable in the equation): Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side. Solve the single variable equation: Use properties of equality to rewrite the equation as an equivalent equation with the variable on one side of the equal sign and a number on the other side.

x + 1 = 5 = 4 = 4 x = - 1 Solve for “x” Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the variable on one side of the equal sign and a number on the other side.

Solve for ‘x’ 4 + 2x + y = 6 ÷ 2 ÷ x + y = 2 2x + y = 2 - y - y 2x = 2 – y 2x = 2 – y Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side.

Solve for “x” yx – 2 = Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side. ÷ y ÷ y yx = 6 yx = 6

Your turn: 17. Solve for ‘k’ 18. Solve for ‘k’ 19. Solve for ‘k’

Vocabulary Formula: An equation that relates two or more quantities, usually represented by variables. Quantity: An measure of a real world physical property (length, width, temperature, pressure, weight, mass, etc.).

Your turn: for the area of a triangle formula: (‘A’ is a function of ‘b’ and ‘h’.) 20. Solve for “b” 21. Solve for “h”. We call this new version of the formula “b” is a function of “h” and “A” “b” is a function of “h” and “A” 22. What do you call this new version of the formula? (similar to: ‘A’ is a function of ‘b’ and ‘h’.)

Your turn: 23. The width of a rectangle is 2 feet. The length is twice the width. What is the perimeter of the rectangle? 24. The width of a rectangle is 3 feet. The length is four times the width. What is the area of the rectangle?

Formulas are used extensively in science. Science and math come together when mathematical equations are used to describe the physical world. Once a formula is known then scientists can use the equation to predict the value of unknown variables in the formula.

Solve for radius We will now solve for “r” In this form, we say that ‘c’ is a function of ‘r’. ‘c’ is a function of ‘r’. ÷ 2π ÷ 2π In this form, we say that ‘r’ is a function of ‘c’. ‘r’ is a function of ‘c’.

Your Turn: 25. Solve for ‘h’. 26. Solve for (Area of a trapezoid: where the length of the parallel bases are of the parallel bases are and the distance between them is ‘h’.) and the distance between them is ‘h’.)

What if two terms have the variable you’re trying to solve for? Solve the equation for “y”. Use “reverse distributive property 9y + 6xy = 30 ÷(9 + 6x) ÷ (9 + 6x) What is “common” to both of the left side terms? the left side terms? “Factor out” the common term “same thing left/right”

Example Solve for ‘x’. ‘x’ is common to both terms  factor it out (reverse distributive property). factor it out (reverse distributive property). How do you turn (3y – 2) into a “one” so that it disappears a “one” so that it disappears on the left side of the equation? on the left side of the equation? ÷(3y – 2) ÷(3y – 2)

27. Solve for ‘x’. Your turn: Your turn: 28. Solve for ‘y’.

Solving formula Problems The perimeter of a rectangular back yard is 41 feet. Its length is 12 feet. What is its width? Draw the picture Write the formula Replace known variables in the formula with constants ÷2 ÷2 Solve for the variable

Solving formula Problems 1. Draw the picture (it helps to see it) 2. Write the formula 3. Replace known variables in the formula with constants 4. Solve for the variable

29. If the base of a triangle is 4 inches and its area is 15 square inches, what is its height? 31. The perimeter of a rectangle is 100 miles. It is 22 miles long. How wide is the rectangle? Your turn: Your turn: 30. The area of a trapezoid is 40 square feet. The length of one base is 8 feet and its height is 3 feet, what is the length of the other base?