Problem 24 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 2 x+7=x-10 $$

Step-by-Step Solution

Answer

The solution is \( x = -17 \).

Step 1: Simplify Both Sides

The equation given is \( 2x + 7 = x - 10 \). Simplifying means rearranging the equation to make it easier to solve. However, both sides of the equation are already simplified.

Step 2: Move Terms Across the Equation

To solve for \( x \), you need all terms involving \( x \) on one side of the equation. Subtract \( x \) from both sides to get: \( 2x - x + 7 = -10 \), which simplifies to \( x + 7 = -10 \).

Step 3: Isolate the Variable

Next, get \( x \) by itself by subtracting 7 from both sides: \( x + 7 - 7 = -10 - 7 \). This simplifies to \( x = -17 \).

Step 4: Check the Solution

Substitute \( x = -17 \) back into the original equation to verify: \( 2(-17) + 7 = -17 - 10 \). This simplifies to \( -34 + 7 = -27 \) and the right side is \( -27 \). Both sides equal \( -27 \), confirming the solution is correct.

Key Concepts

Simplifying Equations

The process of simplifying equations is like cleaning up a cluttered room. You make it easier to see what you really need! When solving linear equations, we start by simplifying each side if possible. In our example equation \(2x + 7 = x - 10\), both sides are already as simple as they can be.
But it's essential to always start by checking for any like terms that can be combined. Like terms are terms that have the same variable with the same exponent.
For instance, if we had \(3x + 2x + 5\), this could be simplified to \(5x + 5\) by combining the \(3x\) and \(2x\). This step makes the solving process smoother by reducing the complexity of the equation. If both sides are simplified, we can proceed with solving the equation which involves getting all terms with \(x\) on one side.

Isolating Variables

Isolating the variable is a crucial step in solving equations. It's like finding a missing piece to complete your puzzle. Your goal is to get the variable \(x\) all alone on one side of the equation.
Let's look at our equation: \(x + 7 = -10\). Here, we want \(x\) by itself, so we eliminate any numbers added to it.
Start by performing the opposite mathematical operation to both sides. For instance, since \(7\) is added to \(x\), subtract \(7\) from both sides: \[ x + 7 - 7 = -10 - 7 \] By doing this, the \(7\) on the left cancels out, leaving us with \(x = -17\).
This method ensures that the integrity of the equation is maintained while steadily zeroing in on \(x\). The equation is now simplified to show \(x\)'s exact value, bringing us closer to verifying our solution.

Checking Solutions

Verifying your solution is vital. Just like checking your work or calculating change at a store. After isolating the variable and solving for \(x\), it's important to substitute your answer back into the original equation to ensure accuracy.
This double-checks that no mistake was made along the way. Let's take \(x = -17\) and insert it back in the equation \(2x + 7 = x - 10\). Substituting gives: \[2(-17) + 7 = -17 - 10\] Simplifying both sides, we get: \[-34 + 7 = -27\] and the right side simplifies to \(-27\). If the left-hand side equals the right-hand side of the original equation, you've found the correct solution. This step reassures you that your math work is solid and your solution is accurate.