Problem 86 - Prealgebra

Question

Translate each of the following into an equation, and then solve the equation. The sum of 8 and 5 is equal to the difference of \(x\) and 7.

Step-by-Step Solution

Answer

The value of \(x\) is 20.

Step 1: Translate the Problem into an Equation

According to the problem, 'the sum of 8 and 5' can be expressed as \(8 + 5\). Also, 'the difference of \(x\) and 7' can be expressed as \(x - 7\). So, the given statement becomes: \(8 + 5 = x - 7\).

Step 2: Simplify the Equation

Simplify the left side of the equation by adding 8 and 5. This gives: \(13 = x - 7\).

Step 3: Solve for \(x\)

To solve for \(x\), you need to isolate \(x\) on one side of the equation. Add 7 to both sides of the equation: \[ 13 + 7 = x - 7 + 7 \]This simplifies to: \(20 = x\). Thus, \(x = 20\).

Key Concepts

Solving Equations

When we're talking about solving equations, we mean finding the value of a variable that makes the equation true. An equation is like a balance scale, what's on one side must match the other. For example, if you have an equation like \(13 = x - 7\), your job is to figure out what number can replace \(x\) to make the statement true.
This involves a few techniques, such as adding, subtracting, multiplying, or dividing both sides by the same number, similar to adjusting both sides of a balance scale to keep them equal. In the example \(13 = x - 7\), we want to isolate \(x\). This means getting \(x\) by itself on one side of the equation. By adding 7 to both sides, we have \(20 = x\). So, \(x\) equals 20.

Translating Word Problems into Equations

Translating word problems into equations might feel like learning a new language, but with a bit of practice, it becomes simple. Start by reading the problem carefully and identifying the quantities involved or relationships described.

• Identify keywords or phrases that suggest mathematical operations: 'sum' means addition, and 'difference' indicates subtraction.
• Assign variables (like \(x\)) to unknown quantities.

Take our example: "The sum of 8 and 5 is equal to the difference of \(x\) and 7." We're told to take the sum (add) 8 and 5, and this equals ("is equal to") the difference (subtract) of \(x\) and 7. By translating these relationships into mathematical terms, we got \(8 + 5 = x - 7\). This is the magic step that turns words into numbers we can solve.

Basic Algebra Operations

Basic algebra operations are the building blocks of all math work you'll encounter. These operations include addition, subtraction, multiplication, and division, and they follow the same basic rules you learned in arithmetic.

• Addition and subtraction help to combine or separate values.
• Multiplication and division work to scale numbers up or down.

You can think of these tasks like rearranging elements to solve a puzzle. In algebra, these operations are often applied to both sides of an equation to maintain balance. For instance, if you're given \(13 + 7 = x - 7 + 7\), adding 7 to both sides cancels the \(-7\) on the right side, isolating \(x\). This strategy reveals \(x = 20\). Once you're comfortable with manipulating these operations, algebra becomes a more logical and satisfying challenge.