Linear Equations: Videos & Practice Problems

Linear equations, formed by adding an equal sign to linear expressions, can be solved using operations like addition, subtraction, multiplication, and division to isolate the variable. Equations can have one solution (conditional), infinite solutions (identity), or no solutions (inconsistent). For example, solving $2x+4=10$ yields $x=3$. Understanding these categories enhances problem-solving skills in algebra.

Introduction to Solving Linear Equtions

In this chapter, we explore the concept of linear equations, which are formed by setting a linear expression equal to a value. For example, the expression \(2x + 3\) becomes a linear equation when we state that it equals 5, written as \(2x + 3 = 5\). The goal is to find the value of \(x\) that satisfies this equation.

To solve linear equations, we utilize various mathematical operations—addition, subtraction, multiplication, and division—to isolate \(x\). It is crucial to perform the same operation on both sides of the equation to maintain equality. For instance, if we have the equation \(x + 2 = 0\), we can isolate \(x\) by subtracting 2 from both sides, resulting in \(x = -2\). Similarly, for the equation \(3x = 12\), we divide both sides by 3 to find \(x = 4\).

When dealing with more complex equations, such as \(2(x - 3) = 0\), we follow a systematic approach. First, we distribute the constant: \(2x - 6 = 0\). Next, we isolate the terms involving \(x\) on one side and constants on the other. By adding 6 to both sides, we get \(2x = 6\). Finally, we divide both sides by 2 to find \(x = 3\). This value is referred to as the solution or root of the equation.

To ensure our solution is correct, we can substitute \(x\) back into the original equation. For example, substituting \(x = 3\) into \(2x - 3 = 0\) gives us \(2(3) - 3 = 0\), which simplifies to \(6 - 3 = 0\), confirming that our solution is valid.

Understanding how to manipulate and solve linear equations is essential, as it lays the groundwork for more advanced mathematical concepts. Mastery of these skills will enable you to tackle a variety of equations effectively.

Solve the Equation. $3\left(2-5x\right)=4x+25$

• A. $x=-27$
• B. $x=-1$
• C. $x=1$
• D. $x=16$

Answer: (B)

Solving Linear Equations with Fractions

When solving linear equations that contain fractions, the first step is to eliminate those fractions to simplify the equation. This can be achieved by using the least common denominator (LCD). For example, consider the equation:

\(\frac{1}{4}x + 2 - \frac{1}{3}x = \frac{1}{12}\).

In this case, the denominators are 4, 3, and 12, and the least common denominator is 12. To eliminate the fractions, multiply the entire equation by 12:

\(12 \left(\frac{1}{4}x + 2\right) - 12 \left(\frac{1}{3}x\right) = 12 \left(\frac{1}{12}\right)\).

This simplifies to:

\(3x + 6 - 4x = 1\).

Now that the fractions are gone, the next step is to distribute any constants. Here, the 3 is distributed to both \(x\) and 2, resulting in:

\(3x + 6 - 4x = 1\).

Next, combine like terms. The terms \(3x\) and \(-4x\) combine to give \(-1x\), leading to:

\(-1x + 6 = 1\).

Now, isolate the variable \(x\) by moving the constant to the other side. Subtract 6 from both sides:

\(-1x = 1 - 6\), which simplifies to \(-1x = -5\).

To solve for \(x\), divide both sides by -1:

\(x = \frac{-5}{-1} = 5\).

Finally, it’s a good practice to check your solution by substituting \(x\) back into the original equation. While this step can be optional, it ensures that your solution is correct, especially when fractions are involved.

Solve the equation.
$\frac92+\frac14\left(x+2\right)=\frac34x$

• A. $x=10$
• B. $x=4$
• C. $x=8$
• D. $x=-1$

Answer: (A)

Categorizing Linear Equations

Linear equations can be categorized based on the number of solutions they possess, which can be expressed as a solution set using set notation. There are three main categories: conditional equations, identity equations, and inconsistent equations.

To illustrate, consider the linear equation 2x + 4 = 10. The first step in solving this equation is to isolate the variable. By subtracting 4 from both sides, we simplify it to 2x = 6. Dividing both sides by 2 gives us x = 3. This equation has a single solution, represented as the solution set {3}. Since it is only true when x equals 3, it is classified as a conditional linear equation.

Next, examine the equation x + 5 = x + 2 + 3. Simplifying the right side results in x + 5 = x + 5. By subtracting x from both sides, we arrive at 0 = 0, a universally true statement. This indicates that any value for x will satisfy the equation, leading to an infinite number of solutions. Thus, the solution set is all real numbers, denoted by ℝ. This type of equation is known as an identity equation because it holds true for all values of x.

Finally, consider the equation x = x + 4. By subtracting x from both sides, we find 0 = 4, which is a false statement. This indicates that there are no values for x that can satisfy the equation, resulting in an empty solution set, represented as {∅}. This scenario is classified as an inconsistent equation, also referred to as a contradiction, as it leads to an impossible statement.

In summary, linear equations can be categorized into three types based on their solutions: conditional equations (one solution), identity equations (infinite solutions), and inconsistent equations (no solutions). Understanding these categories helps in analyzing and solving linear equations effectively.

Solve the equation. Then state whether it is an identity, conditional, or inconsistent equation. 
$5x+17=8x+12-3(x+4)$

• A. Identity
• B. Conditional
• C. Inconsistent

Answer: (C)

Solve the equation. Then state whether it is an identity, conditional, or inconsistent equation. $\frac{x}{4}+\frac16=\frac{x}{3}$

• A. Identity 
• B. Conditional 
• C. Inconsistent 

Answer: (B)

Solve the equation. Then state whether it is an identity, conditional, or inconsistent equation. 
$-2\left(5-3x\right)+x=7x-10$

• A. Identity 
• B. Conditional 
• C. Inconsistent 

Answer: (A)

Here’s what students ask on this topic:

What is a linear equation and how do you solve it?

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form $\mathrm{ax}+b=c$, where $a$, $b$, and $c$ are constants. To solve a linear equation, the goal is to isolate the variable on one side of the equation. This is done by performing operations such as addition, subtraction, multiplication, and division on both sides of the equation. For example, to solve $\mathrm{2x}+3=5$, you would subtract 3 from both sides and then divide by 2 to find $x=1$.

How do you solve linear equations with fractions?

To solve linear equations with fractions, the first step is to eliminate the fractions by finding the least common denominator (LCD) of all the fractions involved. Multiply every term in the equation by the LCD to clear the fractions. For example, in the equation $\frac{1}{4}x+\frac{1}{3}x=\frac{1}{12}$, the LCD is 12. Multiply each term by 12 to get $\mathrm{3x}+\mathrm{4x}=1$. Then, combine like terms and solve for the variable using standard techniques for linear equations.

What are the different types of solutions for linear equations?

Linear equations can have three types of solutions: conditional, identity, and inconsistent. A conditional equation has exactly one solution, meaning there is a specific value for the variable that satisfies the equation. An identity equation has an infinite number of solutions, meaning any value for the variable will satisfy the equation. An inconsistent equation has no solution, meaning there is no value for the variable that will satisfy the equation. These solutions can be expressed using set notation, where a conditional solution is a single number, an identity solution is all real numbers, and an inconsistent solution is the empty set.

How do you verify the solution of a linear equation?

To verify the solution of a linear equation, substitute the solution back into the original equation to check if it satisfies the equation. For example, if you solve the equation $\mathrm{2x}+3=5$ and find $x=1$, substitute $x=1$ back into the equation: $2\times 1+3=5$. Simplifying gives $5=5$, which is true, confirming that $x=1$ is the correct solution.

What is the importance of the least common denominator in solving equations with fractions?

The least common denominator (LCD) is crucial in solving equations with fractions because it allows you to eliminate the fractions, simplifying the equation to a more manageable form. By multiplying every term in the equation by the LCD, you convert the equation into one without fractions, making it easier to apply standard algebraic techniques to solve for the variable. This step is essential for maintaining the equation's balance and ensuring that the solution process is straightforward and efficient.