Problem 52 - Algebra 1

Question

Solve the equation. $$ -(3-x)=-7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-4\).

Step 1: Distribute the negative sign

Distribute the negative sign to each term inside the parentheses to get: \(-(3-x) = -7 \rightarrow (-1)*3-(-1)*x=-7 \rightarrow -3+x=-7 \)

Step 2: Isolate variable x

Move -3 to the other side by adding 3 to both sides to isolate x: \( -3+x=-7 \rightarrow -3+x+3=-7+3 \rightarrow x=-4 \)

Key Concepts

Distributive Property

To solve linear equations, the distributive property is a powerful tool. When you see a negative sign outside parentheses, it means you need to distribute this negative sign to each term inside. In our example, we have

• \(-(3-x)\).

This can be rewritten as

• \(-1 \times 3\) and \(-1 \times (-x)\).

So:

• \( -3+x \)

By distributing, you transform complex expressions into simpler forms. Always remember to multiply every term inside the parentheses.

Isolation of Variable

The main goal in solving linear equations is to isolate the variable. Isolation means getting the variable, like \(x\), alone on one side of the equation. In the equation:

• \(-3 + x = -7\)

you want to move \(-3\) to the other side. This is done by performing the opposite operation—in this case, adding 3—to both sides of the equation. So, you add 3:

• \(-3 + x + 3 = -7 + 3\)

Simplifying gives:

• \(x = -4\)

Isolating the variable

• lets you solve for its value easily.

Integer Operations

Understanding integer operations is crucial for solving equations accurately. Here, it's important to handle positive and negative numbers correctly. In the transition

• from \(-3 + x = -7\)

to

• \(x = -4\)

You need to add integers. When adding \(-7 + 3\), consider the signs:

• Moving left three steps from \(-7\) lands you at \(-4\).

When managing integers:

• Add or subtract carefully and watch for incorrect sign operations.

This process ensures you maintain accurate calculations throughout.