Problem 124 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 5 x-2 x+6 x=13 $$

Step-by-Step Solution

Answer

\(x = \frac{13}{9}\).

Step 1: Combine Like Terms

The equation is given as: \(5x - 2x + 6x = 13\). First, combine the like terms on the left side of the equation. Like terms are terms with the same variable. Here, \(5x\), \(-2x\), and \(6x\) are like terms. Thus, you sum them: \[(5x - 2x + 6x) = 9x\]Thus, the equation simplifies to \(9x = 13\).

Step 2: Isolate the Variable

To solve for \(x\), you need to isolate it on one side of the equation. Here, \(9x = 13\). Divide both sides by 9 to solve for \(x\):\[x = \frac{13}{9}\]

Key Concepts

Combine Like Terms

When solving linear equations, one of the first steps is often to combine like terms. Like terms are terms that have the exact same variable part. This means they share the same variable raised to the same exponent. In our example, the original equation is:

• \(5x - 2x + 6x = 13\)

The like terms here are \(5x\), \(-2x\), and \(6x\). These terms can be added together because they all contain the variable \(x\). The process involves adding or subtracting the coefficients (the numerical parts in front of the variables) while keeping the variable part unchanged.

When combined, these like terms simplify to:

• \((5 - 2 + 6)x = 9x\)

Combining like terms helps you reduce the equation to a simpler form, making it easier to solve. Remember, the key is to ensure that all terms involved have the same variable with the same power.

Isolate the Variable

After combining like terms in an equation, the next crucial step is to isolate the variable. This means getting the variable, in this case \(x\), alone on one side of the equation. Let's look at our equation after combining like terms:

• \(9x = 13\)

To isolate \(x\), you need to get rid of the coefficient next to it. In this example, \(9\) is the coefficient, which means \(x\) is being multiplied by \(9\). To counter this, divide both sides of the equation by \(9\):

• \(x = \frac{13}{9}\)

This operation allows \(x\) to stand alone on one side of the equation, which is precisely what isolating the variable means. It's important to perform the same operation on both sides to maintain equality. This step is fundamental in algebra to find the value of unknowns.

Simplify Equations

Simplifying equations goes hand-in-hand with the previous steps and is essential for making complex equations more manageable. In the process of simplifying, you are essentially cleaning up the equation to its most straightforward form. This usually involves:

• Combining like terms
• Applying arithmetic operations
• Reducing fractions if necessary

In our example, after combining like terms and isolating the variable, we reached:

• \(x = \frac{13}{9}\)

At this stage, the equation is fully simplified. However, if your answer were something like \(\frac{14}{10}\), you would further simplify it to \(\frac{7}{5}\) by dividing the numerator and the denominator by their greatest common divisor. Simplification makes the equation easier to understand and communicate, which is why it's an essential step in solving equations.