Chapter 2
Linear Equations
Students will be able to translate sentence into equations and vice versa
2-1 Writing Equations
• To write an equation, identify the unknown for which you are looking
and assign a variable to it. Then write the sentence as an equation.
Look for key words such as is, is as much as, is the same as, or is
identical to, that indicate where you should place the equals sign.
• Seven times a number squared is five times the difference of k and m
2
• 7
*
x
= 5 *
(k-m)
• 7x 2 =5(k-m)
Writing verbal equations
• 15=25u2 + 2
• Fifteen is the sum of twenty five times a number squared and two
• 6x – 15 = 45
• Fifteen less than 6 times a number is equivalent to 45
Students will be able to solve equations using addition, subtraction, multiplication and division.
2.2 Solving One Step Equations
• Understanding equality. If I tell you 12 + x = 36, then theoretically if I add or
subtract the same value to both sides of an equal sign, the equality remains
true. Hence (-12) + 12 + x = 36 + (-12) : x = 24.
• Same is true for multiplication and division. If 3x=12 then I can divide both
sides by 3, hence 3x÷3=12÷3 : x=4
• Goal – Isolate x on one side of the equation by doing the opposite of what is
being done to the x initially.
• In the first example 12 was being added to x, now we want to subtract 12 from both
sides
• In the 2nd example x is being multiplied by 3, hence we want to divide both sides by 3.
2.3 Solving Multi-Step Equations
• Term – Algebraic expressions separated by + or -.
• Let us consider: 3x – 7 = -10
• Terms: {3x, -7, -10}. Since the term -7 is on the same side of the equal sign
as the x, we need to move it to the other side first, by adding 7 to both sides
• 3x – 7 +7 = -10 + 7 : 3x = -3
• Now we would divide both sides by 3 to get x = -1
HINT: When you are isolating x, generally speaking (not always true), you will be
using PEMDAS in reserve order. You’ll add or subtract terms not involving x (usually
constants), then dividing by the coefficient of x or multiplying by what x is being
divided by, then you’d worry about exponents.
Students will be able to solve equations with the variable on either side of an equal sign.
2.4 Solving Equations with the Variable on
Each Side.
• Let us consider 2 + 5x = 3x – 6
• The first step is to put the terms 5x and 3x on the same side. You can put
them together on either side.
• You are NOT combining like terms, you are moving such terms
• 2 + 5x – 3x = 3x – 6 – 3x : 2 + 2x = -6
• Now we can move the 2 and get 2x = -8: x=-4
Students will learn to solve equations with grouping symbols.
2-4 Solving equations with grouping symbols
• 2 methods. Let us consider 4x = 2(x-3)
• Distribution: 4x = 2x – 6: From this point we can say 4x - 2x = 2x - 2x – 6
• 2x = -6: x = -3
• Divide by the coefficient of the grouping if possible and convenient.
•
4𝑥
2
=
2(𝑥−3)
;
2
2x = x-3; Then we can subtract x from both sides
• 2x – x = x – 3 – x; x = -3
Students will be able to solve and evaluate equations involving absolute value.
2.5 Solving Equations Involving Absolute Value
• Absolute Value – Anything inside the absolute value lines |x| takes on
the positive value of what is inside. |5| = 5 also |-5| = 5
• Evaluate |5 – 9|: |-4| = 4
• Solve for x: |-2x + 6| = 8
• Your first instinct is to say -2x + 6 = 8: -2x = 2: x = -1 And this is true. However, |-2x+6|
can also take on another value, specifically -8. Remember, |-8| = 8 (the desired result)
• Your 2nd solution is to set the equation equal to -8. -2x + 6 = -8: -2x = -14: x = 7
• FINAL Answer. X = {-1, 7}