Problem 53 - Algebra A Combined Function

Question

\(6 x+10=-20\)

Step-by-Step Solution

Answer

The solution is \( x = -5 \).

Step 1: Subtract 10 from Both Sides

To isolate the term with the variable on one side, we first need to eliminate the constant term on the left side of the equation. We do this by subtracting 10 from both sides: \[ 6x + 10 - 10 = -20 - 10 \] which simplifies to: \[ 6x = -30 \]

Step 2: Divide Both Sides by 6

Now, to solve for \( x \), we need to get rid of the 6 that is being multiplied by \( x \). This is done by dividing both sides of the equation by 6: \[ \frac{6x}{6} = \frac{-30}{6} \]which simplifies to: \[ x = -5 \]

Key Concepts

Solving Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true. It's like a balance scale, where you aim to maintain equilibrium while discovering what the variable is. There are a few straightforward steps to follow:

• Identify the terms on each side of the equation. Look for constants and coefficients multiplied by variables.
• The goal is to get the variable by itself on one side of the equation, and a number on the other side.
• To do this, you will perform inverse operations until the variable is isolated.

In our example, the equation we start with is:\[ 6x + 10 = -20 \].
You begin by performing operations that help separate the variable term, which leads us to the next crucial concept.

Isolation of Variables

Isolation of variables is a key step in solving equations. This part usually comes after you've rearranged your equation so it involves the term with the unknown variable on one side.

• Begin by isolating the variable term. In our case, \( 6x \) is the variable term.
• Identify the constants on the same side as your variable, and eliminate them by using inverse operations.
• This could mean subtracting a constant on one side if it was added or adding a constant if it was subtracted.

In the example \( 6x + 10 = -20 \), we isolate the variable by eliminating the '10' from the left side. We do this by subtracting 10 from both sides, yielding:\[ 6x = -30 \].
This manipulation sets the stage for simplifying the equation.

Simplifying Equations

Simplifying equations is often the final step that leads to the solution. By this time, you've isolated the variable, and now it's about determining its actual value.

• With \( 6x = -30 \), the next step is to remove the coefficient "6" from the 'x'.
• Since 6 is multiplied by \( x \), use division to cancel it out.
• Divide both sides of the equation by 6 to keep it balanced, resulting in \( x = -5 \).

This division simplifies the equation to reveal the value of \( x \).
Successfully simplifying the equation confirms that your operations have maintained equality, and thus the solution \( x = -5 \) is correct.