Problem 70 - Algebra 1

Question

Solve the equation. $$ 5 x-7+x=19 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4.33333\).

Step 1: Combine like terms

In this case, we can combine the \(x\) terms on the left hand side of the equation. \(5x + x = 6x\). So, the equation now looks like this: \(6x - 7 = 19\).

Step 2: Isolate the variable term

To do this, add 7 to both sides of the equation to get rid of the -7 on the left side. This will yield the equation \(6x = 26\).

Step 3: Solve for 'x'

Finally, divide both sides of the equation by 6 to solve for \(x\). So, \(x = 26 / 6\) or \(x = 4.33333\).

Key Concepts

Combining Like Terms

When we talk about "combining like terms," we're discussing the process of simplifying expressions in an equation. Terms are considered "like" if they have the same variables raised to the same power. In the provided equation, we have two x terms: \(5x\) and \(x\). Since both terms have the variable \(x\), we can combine them.

• Add the coefficients: The coefficients are numbers multiplying the variables. In this case, you add the coefficients of \(5x\) and \(x\) together: \(5 + 1 = 6\).
• Rewrite the expression: Combining \(5x\) and \(x\) gives you \(6x\).

By combining like terms, our equation becomes simpler and easier to work with: \(6x - 7 = 19\). By simplifying expressions before solving, you'll find the entire process more straightforward.

Isolation of Variables

Isolating the variable is a key step in solving linear equations. The aim here is to have the variable, \(x\) in this case, all by itself on one side of the equation. This step allows us to better see what \(x\) equals.

• Addition or Subtraction: First, look at the equation \(6x - 7 = 19\). We want to eliminate \(-7\) from the left side to isolate the terms with \(x\). To do this, add \(7\) to both sides of the equation, effectively canceling \(-7\) out.
• New Equation: After adding \(7\) to both sides, the equation transforms to \(6x = 26\).

By performing this operation, we have isolated the \(6x\) on one side and are one step closer to finding out what \(x\) is.

Solving for x

The final step in solving the equation is to solve for \(x\). Our equation is already simplified to \(6x = 26\), thanks to the previous steps.

• Division: To solve for \(x\), divide both sides of the equation by \(6\). This will effectively "cancel" the \(6\), leaving \(x\) alone on one side.
• Calculation: Doing the math, \(x = \frac{26}{6}\), which simplifies to \(x = 4.33333\). Another way to view this number is in fraction form, \(x = \frac{13}{3}\).

This division step gives you the exact value of \(x\), completing the process of solving the equation. Always remember, the main goal of solving for \(x\) is to find its numerical value, which in this case was a decimal approximation.