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Chapter 2: Problem 27

Solve each linear equation. Show your work and check your answer. $$-2 x+5=7$$

Short Answer

Expert verified
x = -1

Step by step solution

01

Isolate the variable term

First, move the constant term to the other side of the equation by subtracting 5 from both sides. -2x + 5 - 5 = 7 - 5 This simplifies to: -2x = 2
02

Solve for the variable

Next, solve for x by dividing both sides of the equation by -2. \(-2x \div -2 = 2 \div -2\) This simplifies to: x = -1
03

Check the solution

Substitute x = -1 back into the original equation to verify that it satisfies the equation. -2(-1) + 5 = 7 This simplifies to: 2 + 5 = 7 Since both sides of the equation are equal, x = -1 is the correct solution.

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Key Concepts

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

Isolate the variable
When solving a linear equation, the first step is to isolate the variable. This means you want to get the variable (like 'x') all by itself on one side of the equation. Take the equation \(-2x + 5 = 7\). Notice that the variable term \(-2x\) is not alone. We need to move the constant term (which is a number without a variable) to the other side.
To do this, subtract 5 from both sides of the equation:
\(-2x + 5 - 5 = 7 - 5\).
This simplifies to:
\(-2x = 2\).
Now, our variable term is isolated. We're ready to solve for the variable!
Equation solving steps
Once the variable is isolated, you can solve for it. Take the isolated equation \(-2x = 2\). You'll get the variable by itself by undoing any multiplication or division present. Here, we have \(-2\) times \(x\).
To undo this multiplication, divide both sides by \(-2\):
\(-2x \div -2 = 2 \div -2\).
This simplifies to:
\(x = -1\).
At this point, you've solved the equation using a step-by-step approach:
  • Isolate the variable term by moving the constant.
  • Solve for the variable by performing necessary operations.
This helps you systematically find the solution.
Checking solutions
After solving the equation, it's essential to check the solution to ensure it is correct. You do this by substituting the found value back into the original equation to see if it satisfies the equation. Take your solution \(x = -1\). Substitute it back into the original equation:
\(-2(-1) + 5 = 7\).
Simplifying this, you'll get:
\(+2 + 5 = 7\).
Both sides equal 7, so the equation holds true.
This means \(x = -1\) is indeed the correct solution. Always verify by substituting your solution back into the equation to catch any mistakes early!

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