Solving Linear Equations with Rational Coefficients
1. • Students extend the use of the properties of
equality to solve linear equations having
rational coefficients.
Unit 2: Lesson 5
Solving a Linear Equation
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5. To solve an equation means to find all the numbers x, if they exist, so
that the given equation is true.
In some cases, some simple guess work can lead us to a solution.
For example, consider the following equation:
4x + 1 = 13
What number x would make this equation true? That is, what value
of x would make the left side equal to the right side?
In other cases, guessing the correct answer is not so easy. Consider
the following equation:
3(4x – 9) + 10 = 15x + 2 + 7x
Can you guess a number for x that would make this equation true????
6. Guessing is not always an efficient strategy for solving equations. In
the last example, there are several terms in each of the linear
expressions comprising the equation. This makes it more difficult to
easily guess a solution.
For this reason, we want to use what we know about the properties
of equality to transform equations into equations with fewer terms.
The ultimate goal of solving any equation is to get it into the form of x
(or whatever symbol is being used in the equation) equal to a
constant.
9. There are four properties of equality that will allow us to
transform an equation into the form we want.
If A,B, and C are any rational numbers, then
• If A = B, then A + C = B + C.
• If A = B, then A − C = B − C.
• If A = B, then A ⋅ C = B ⋅ C.
• If A = B, then
𝐴
𝐶
=
𝐵
𝐶
, where C is not equal to zero.
Properties of Equality
10. All four of the properties require us to start off with A = B.
That is, we have to assume that a given equation has an
expression on the left side that is equal to the expression
on the right side.
Working under that assumption, each time we use one of
the properties of equality, we are transforming the
equation into another equation that is also true, i.e., left
side equals right side.
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23. The properties of equality are not the only properties we
can use with equations. What other properties do we
know that could make solving an equation more efficient?
We know the distributive property, which allows us to
expand and simplify expressions.
We know the commutative and associative properties,
which allow us to rearrange and group terms within
expressions.
24. Now we will solve the linear equation
1
5
x + 13 + x = 1 − 9x + 22
25.
26. Wrap Up
We know that properties of equality and inverse operations,
when used to transform equations, make equations with
fewer terms that are simpler to solve.
When solving an equation, we want the answer to be in the
form of the symbol x equal to a constant.
What now???
• BSN Lesson – Solving One Step Equations
• IXL Skill U.5 – Solve One-Step Equations
• BSN Lesson – Solving Two- Step Equations
• IXL Skill U.6 – Solve Two-Step Equations
