/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Solve the equation. $$ -(3-x... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 52

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

Short Answer

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

Step by step solution

01

Distribute the negative sign

Distribute the negative sign to each term inside the parentheses to get: \(-(3-x) = -7 \rightarrow (-1)*3-(-1)*x=-7 \rightarrow -3+x=-7 \)
02

Isolate variable x

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊n this case, adding 3鈥攖o 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.

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