Linear Equations in One Variable

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Presentation transcript:

Linear Equations in One Variable

Basic Ideas and Definitions An equation is a sentence that expresses the equality of two algebraic expressions.

Basic Ideas and Definitions An equation is a sentence that expresses the equality of two algebraic expressions. Consider the equation 2x+1=7

Basic Ideas and Definitions An equation is a sentence that expresses the equality of two algebraic expressions. Consider the equation 2x+1=7 We say 2(3) +1=7 is true.

Basic Ideas and Definitions An equation is a sentence that expresses the equality of two algebraic expressions. Consider the equation 2x+1=7 We say 2(3) +1=7 is true. We say that 3 satisfies the equation.

Basic Ideas and Definitions Algebraic Expressions 8x + 9 y - 4

Basic Ideas and Definitions A Linear Equation in one variable involves only real numbers and one variable raised to the first power x + 1 = -2 x - 3 = 5 2k + 5 = 10

Basic Ideas and Definitions A Linear Equation in one variable can be written in the form Ax + B = C

Basic Ideas and Definitions Any number that satisfies the equation is called a solution or root to the equation.

Solution Set The set of all solutions to an equation is called the solution set to the equation.

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Determine whether each equation is satisfied by the number following Example 1 Determine whether each equation is satisfied by the number following the equation

Solving Equations To solve an equation means to find its solution set. The most basic method for solving equations involves the properties of equality.

Properties of Equality Addition-Subtraction Property of Equality The same real number may be added to or subtracted from each side of an equation without changing the solution set.

Properties of Equality Multiplication-Division Property of Equality Each side of an equation may be multiplied by or divided by he same real number without changing the solution set.

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the equation 4x - 2x - 5 = 4 + 6x + 3 Example 1 Solve the equation 4x - 2x - 5 = 4 + 6x + 3

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Solve the Equation 2(k - 5) + 3k = k + 6 Example 2 Solve the Equation 2(k - 5) + 3k = k + 6

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

Example 3 Solve the Equation

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

.06x + .09(15-x)=.07(15) Example 4 Solve

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x-9 = 4(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-3)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4)

Example 5 Solve 5x - 15 = 5(x-4) Contradiction

Example 5 Solve 5x - 15 = 5(x-4) Contradiction

Identity, Conditional Equation, Contradiction An identity is an equation that is satisfied by every number for which both sides are defined.

Identity, Conditional Equation, Contradiction A conditional equation is an equation that is satisfied by at least one number but is not an identity.

Identity, Conditional Equation, Contradiction A Contradiction is an equation whose solution is the empty set.

L 1.1 # 1 - 82 Every Other Odd Problem 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, etc.