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Lesson Explainer: Solving Linear Equations over the Real Numbers Mathematics

In this explainer, we will learn how to solve linear equations over the real numbers and represent their solutions on a number line.

We will be focusing on linear equations, which are also known as first-degree equations, because the exponent of the unknown variable is 1. The equations we will consider will mainly be of the form 𝑎𝑥+𝑏=𝑐, where 𝑎, 𝑏, and 𝑐 are real constants and 𝑎 is nonzero.

Recall that when solving equations, we always perform the same operations to both sides of the equation so that the equation remains balanced. The operations we perform will be the inverses of the operations performed when forming the equation. To solve a two-step linear equation of the form 𝑎𝑥+𝑏=𝑐, we first need to isolate the 𝑥 term and then divide by its coefficient. This is equivalent to adding the additive inverse of 𝑏 to both sides of the equation and then multiplying both sides by the multiplicative inverse of 𝑎. We recap the formal method for solving such an equation and representing the solution on a number line in our first example.

Example 1: Solving a Two-Step Linear Equation and Representing the Solution on a Number Line

  1. Find, in , the solution set of the equation 2𝑥+3=5.
  2. Which of the following represents the solution of the equation on a number line?

Answer

Part 1

In order to solve this equation, we first need to isolate the unknown variable. Subtracting 3 from each side of the equation leads to 2𝑥+33=532𝑥=2.

We then divide each side of the equation by the coefficient of 𝑥, which is 2: 2𝑥2=22𝑥=1.

The solution set of the given equation is {1}.

Part 2

The solution to this equation is a single value (1) and so should be represented on a number line by a solid dot at this value. The correct representation is therefore option A.

The two-step equation we have just considered was reasonably straightforward because the coefficients and constants involved were all integers. More complicated equations may involve coefficients or constant terms that are radicals, leading to solutions that themselves involve radicals. Let us now consider an example of this.

Example 2: Solving a Linear Equation with a Radical Solution

Which of the following is the solution set of the equation 2𝑥+23=2 in ?

  1. 1+3
  2. 13
  3. 2+3
  4. 23
  5. 43

Answer

To isolate the 𝑥 term, we begin by subtracting 23 from each side of the equation: 2𝑥+2323=2232𝑥=223.

We then divide both sides of the equation by 2 (the coefficient of 𝑥): 2𝑥2=2232𝑥=13.

The solution set of this equation is 13, which is option B.

We will now consider an example in which the coefficient of the unknown variable involves a radical. During solving, we will be required to divide by this radical, which will lead to an answer involving an irrational denominator. At this point, we will recall the process for rationalizing the denominator of a fraction. In this example, we will also demonstrate how to represent the solution to an equation on a number line.

Example 3: Solving and Representing the Solution to a Two-Step Equation Involving a Radical Coefficient on a Number Line

  1. Find, in , the solution set of the equation 3𝑥+2=5.
  2. Which of the following shows the solution of the equation on a number line?

Answer

Part 1

We begin by isolating the 𝑥 term. To do this, we subtract 2 from each side of the equation, leading to 3𝑥+22=523𝑥=3.

Next, we need to divide by the coefficient of 𝑥. Here, the coefficient of 𝑥 is a radical, but as this is just a number, there is no issue with dividing by it. Dividing both sides of the equation by 3 gives 3𝑥3=33𝑥=33.

While we have found the value of 𝑥 that solves this equation, this is not our final answer, as the denominator of this fraction is irrational. We proceed by multiplying both the numerator and denominator of this fraction by 3 to give an equivalent fraction with a rational denominator: 𝑥=33×33=333=3.

The solution set of the given equation is 3.

Part 2

To represent this solution on a number line, we need to have some idea of its value. 3 is approximately equal to 1.73, so this value is between 1 and 2 and closer to 2 than 1. The solution is therefore represented by a solid dot in this approximate location:

From the five options given, this is option D.

We summarize the steps involved in solving two-step linear equations below.

How To: Solving a Two-Step Linear Equation of the Form 𝑎𝑥 + 𝑏 = 𝑐

  • Isolate the unknown variable by adding the additive inverse of 𝑏 to both sides of the equation.
  • Divide both sides of the equation by the coefficient of the unknown variable. This is equivalent to multiplying both sides by the multiplicative inverse of 𝑎.
  • If this leads to a fraction with an irrational denominator, rationalize the denominator by multiplying both the numerator and denominator of the fraction by the radical in the denominator.

We now consider one final example in which the equation we are required to solve involves a radical coefficient but also requires some initial simplification.

Example 4: Solving a Linear Equation Involving Radical Coefficients and Constants

Find, in , the solution set of the equation 53𝑥2=45.

Answer

There are two approaches to answering this question. We could begin by distributing the parentheses on the left-hand side. However, as the unknown variable appears inside the parentheses, it will be simpler to begin by dividing both sides of the equation by 5: 53𝑥25=455.

On each side of the equation, the factors of 5 in the numerator and denominator cancel each other out, leading to 3𝑥2=4.

Next, we isolate the unknown variable by adding 2 to each side of the equation: 3𝑥2+2=4+23𝑥=6.

We then divide by the coefficient of 𝑥: 3𝑥3=63𝑥=63.

Finally, we need to rationalize the denominator of this quotient by multiplying both the numerator and denominator by 3: 𝑥=63×33=633=23.

The solution set of the given equation is 23.

If we wished to represent the solution to the previous example on a number line, we would need to approximate the value of 23. 3 is approximately equal to 1.73, and by doubling this, we find that 23 is approximately equal to 3.46, or 3.5 to one decimal place. We would therefore represent this solution by a solid dot halfway between 3 and 4 on a number line:

Let us finish by recapping some key points.

Key Points

  • When solving equations, we always perform the same operations on both sides.
  • A two-step linear equation of the form 𝑎𝑥+𝑏=𝑐 can be solved by first isolating the unknown variable and then dividing by its coefficient.
  • If the coefficient is a radical, this will usually lead to a fractional answer with an irrational denominator. The denominator should be rationalized by multiplying both the numerator and denominator of the fraction by the radical in the denominator and then simplifying.
  • The solution to an equation can be represented using a solid dot on a number line. In the case of irrational solutions, the position of the dot will be approximate.

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