Solving Linear Equations: Videos & Practice Problems

In mathematics, a linear equation is formed when a linear expression, such as \(2x + 3\), is set equal to a value, for example, \(5\). This transforms the expression into an equation that we can solve to find the unknown variable \(x\). The goal is to determine the value of \(x\) that satisfies the equation, making the statement true.

To solve a linear equation, we utilize various operations: addition, subtraction, multiplication, and division. It is crucial to apply these operations to 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 isolate \(x\), yielding \(x = 4\).

When isolating \(x\), we often perform opposite operations. If a number is added to \(x\), we subtract it to eliminate it. Conversely, if \(x\) is multiplied by a number, we divide to isolate \(x\). This principle extends to more complex equations where multiple operations may be necessary. For example, in the equation \(2(x - 3) = 0\), we first distribute the \(2\) to get \(2x - 6 = 0\). Next, we add \(6\) to both sides, resulting in \(2x = 6\), and finally divide by \(2\) to find \(x = 3\).

After solving for \(x\), it is essential to verify the solution by substituting \(x\) back into the original equation. For instance, substituting \(3\) into the equation \(2x - 3 = 0\) confirms that \(0 = 0\), validating our solution. This process of solving and checking ensures that we have accurately determined the value of the variable in the equation.

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:

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

To eliminate the fractions, identify the denominators, which in this case are 4, 3, and 12. The least common denominator is 12. Multiply every term in the 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 eliminated, the next step is to distribute any constants. In this case, the 3 is distributed to both \(x\) and 2, resulting in:

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

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

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

Now, isolate the variable \(x\) by moving the constant term 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. This step, while optional, ensures that your solution is correct. Thus, the solution to the equation is:

\(x = 5\).

Linear equations can be categorized based on the number of solutions they possess, which can be expressed in a solution set using set notation. The three categories of linear equations are 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, which can be 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 moving all terms involving x to one side and constants to the other, we find 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.

Finally, consider the equation x = x + 4. Rearranging gives 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 {∅}. Such equations are termed inconsistent equations, as they lead to contradictions.

In summary, linear equations can be categorized as follows: a conditional equation has one solution, an identity equation has infinitely many solutions, and an inconsistent equation has no solutions. Understanding these categories helps in analyzing and solving linear equations effectively.