*To solve the equation 5x – 17 = -x + 7, first, add x to both sides to simplify: 5x + x – 17 = 7. This becomes 6x – 17 = 7. Next, add 17 to both sides: 6x = 24. Finally, divide both sides by 6: x = 4. So, the solution to the equation is x = 4.*

**Understanding the Equation:**

Let’s start by understanding the equation itself. We have:

5x – 17 = -x + 7

This equation contains one variable, x, and two sides separated by an equal sign. The goal is to find the value of x that makes both sides equal.

**Step 1: Simplify the Equation:**

The first step in solving this equation is to simplify it. To do this, we can start by moving all the x terms to one side of the equation and the constant terms to the other side.

Adding x to both sides:

5x + x – 17 = 7

Now, we have:

6x – 17 = 7

**Step 2: Isolate x:**

The next step is to isolate x on one side of the equation. To do this, we need to get rid of the -17 on the left side. We can achieve this by adding 17 to both sides of the equation:

6x – 17 + 17 = 7 + 17

Simplifying further:

6x = 24

**Step 3: Solve for x:**

Now that we have 6x = 24, we can solve for x by dividing both sides by 6:

(6x) / 6 = 24 / 6

This simplifies to:

x = 4

**The Solution:**

So, the solution to the equation 5x – 17 = -x + 7 is x = 4.

**Interpreting the Solution:**

This means that if we substitute x with 4 in the original equation, both sides will be equal:

5(4) – 17 = -4 + 7

20 – 17 = 3

3 = 3

The equation holds true, verifying that x = 4 is indeed the solution.

**Real-World Applications:**

Linear equations like this one have numerous real-world applications. For example:

**Finance:**Linear equations are used to calculate interest rates, investment returns, and loan payments.**Physics:**They are used to model physical phenomena, such as motion and electricity.**Engineering:**Engineers use linear equations in designing structures, circuits, and systems.**Economics:**Linear equations help economists analyze supply and demand, production, and cost functions.**Statistics:**Linear regression, a statistical technique, uses linear equations to model relationships between variables.

**Conclusion:**

Solving linear equations is a foundational skill in mathematics and has widespread applications in various fields. The process involves simplifying the equation, isolating the variable, and finding the value that makes both sides equal. In the case of 5x – 17 = -x + 7, we found that x = 4 is the solution, demonstrating the power and utility of algebraic techniques in solving real-world problems.

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