3.1 Linear Equations: One Transformation Linear Equations in One Variable A linear equation in one...
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Transcript of 3.1 Linear Equations: One Transformation Linear Equations in One Variable A linear equation in one...
3.1 Linear Equations: One TransformationLinear Equations in One Variable
A linear equation in one variable is an equation that can be written in the form
ax + by = c, where a, b, and c are real numbers and a ≠ 0.An equation is a statement with an equals sign.
3.1 Linear Equations:One Transformation
Original
Equation
Equivalent
Equation
1. Add the same number to both sides. x - 4 = 7 x = 11
2. Subtract the same number to both sides. x + 2 = 9 x = 7
3. Multiply both sides by the same number. x/2 = 4 x = 8
4. Divide both sides by the same number. 5x = 20 x = 4
5. Interchange the two sides. x = 7 7 = x
3.1 Linear Equations:One Transformation Two equations are equivalent if they have the
same solution set. In solving an equation, the goal is to isolate the
variable by using inverse operations.
3.1 Linear Equations:One Transformation
Solve x + 3 = 7 x + 3 - 3 = 7 - 3 Add -3 to both sides x = 4 Simplify
3.1 Linear Equations:One Transformation
Solve 3x = -12
3x
3
12
3 Divide both sides by 3
x 4 Simplify
3.1 Linear Equations:One Transformation
Solve -4 = x - 7
4 + 7 = x - 7 + 7 Add 7 to both sides.
3 x Simplify
x 3 Transpose
3.1 Linear Equations:One TransformationSolve
x
5= -9
x
5
5 = -9(5) Multiply both sides by 5.
x 45 Simplify
3.1 Linear Equations:One Transformation
Properties of Equalities
1. If a = b, then a + c, = b + c Addition Property
2. If a = b, then a - c, = b - c Subtraction Property
3. If a = b, then ac = bc Multiplication Property
4. If a b and c 0, then a
c
b
c Division Property
3.2 Linear Equations: Two or More Transformations Using Two or More Transformations
1. Simplify both sides of the equation.
2. Use inverse operations to isolate the variable.
3. Check the solution.
3.2 Linear Equations:Two or More TransformationsSolve
4x + 7 - 2x =13
2x 7 13 Combine like terms.
2x 6 Subtract 7 from both sides.
x 3 Divide both sides by 2.
3.2 Linear Equations:Two or More TransformationsSolve
5x +2(3 - x) =15
5x 6 2x 15 Remove parentheses
3x 6 15 Combine like terms.
3x 9 Subtract 6 from both sides.
x 3 Divide both sides by 3.
3.2 Linear Equations: Two or More Transformations
Solve: 3x + 6 = 12
3x + 6 - 6 = 12 – 6 Add -6 to both sides
3x = 6 Combine like terms
x = 2 Divide both sides by 3
3.2 Linear Equations: Two or More Transformations
Solve: 4.5 = 3 + 2x
4.5 - 3 = 3 - 3 + 2x Add -3 to both sides
1.5 = 2x Divide both sides by 2
.75 = x
3.2 Linear Equations: Two or More Transformations
Solve: -2x + 2 - 4x = 20
-6x + 2 = 20 Combine like terms
-6x + 2 – 2 = 20 – 2 Add -2 to both sides
-6x = 18 Divide by -6
x = -3
3.2 Linear Equations: Two or More Transformations
Solve: 4(x - 2) = -10
4x - 8 = -10 Remove parenthesis
4x – 8 + 8 = -10 + 8 Add 8 to both sides
4x = -2 Combine like terms
x = -2/4 = -1/2 Divide both sides by -4
3.2 Linear Equations: Two or More Transformations
Solve:
-3(x - 2) = 21
-3x + 6 = 21 Remove parenthesis
-3x + 6 - 6 = -21 - 6 Add -6 to both sides
-3x = -27 Combine like terms
x = 9 Divide both sides by -3
3.3 Solving Equations with Variables on Both Sides
Collect variables on the side with the greatest variable coefficient
3x 5 8x 30
3x 5 3x 8x 30 3x
5 5x 30
35 5x
7 x
3x 5 8x 30
3x 5 8x 8x 30 8x
5x 5 30
5x 35
x 7
3.3 Solving Equations with Variables on Both Sides
Solve:
4(y 2) 6y 2 8y
4y 8 2y 2
4y 8 2y 2y 2 2y
6y 8 2
6y 8 8 2 8
6y 6
y 1
3.3 Solving Equations with Variables on Both Sides
Solve:
3(x 5) 2x 10 4x
3x 15 2x 10
5x 15 10
5x 5
x 1
3.4 Problem SolvingGeneral Strategies for Problem Solving
1. Read and understand the problem.Choose a variableConstruct a diagram
2. Translate the problem into an equation. 3. Solve the equation. 4. Interpret the results.(Check your answer)
3.5 Solving Equations That Involve Decimals
12.3x - 5.1 = 17
Solve:
12.3x - 5.1+5.1 = 17 + 5.1
12.3x = 22.1
x 22.1
12.3
x 1.796747
3.6 Literal Equations
Solve for v d =vt
Solve for l 2w +2l =P
2l =P - 2w
Solve for x y = mx + b
l P 2w
2
y b mx
y b
mx
d
tv
3.7 Scatter Plots
Quadrant IQuadrant II
Quadrant III Quadrant IV
3.7 Scatter Plots
A(3,2)
BB(-1,-3)A
C(-5,0)B