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Chapter 2: Problem 28

Solve each equation. Check all solutions. SEE EXAMPLE 1. (OBJECTIVE 1) $$-8 x+5=21$$

Short Answer

Expert verified
The solution to the equation \(-8x + 5 = 21\) is \(x = -2\)

Step by step solution

01

Simplify the equation by moving constant term to right side

The equation is : \(-8x + 5 = 21\). Add \(8x\) to both sides of the equation to isolate the variable term. This gives us: \(-8x = 21 - 5\).
02

Solve the equation for x

From previous step we have: \(-8x = 16\). To get \(x\), divide both sides of equation by \(-8\). Thus, \(x = 16 / -8\). Therefore, the value of \(x\) is \(-2\).
03

Verify the solution

Substitute \(x = -2\) in the original equation \(-8x + 5 =21\). If the left hand side equals right hand side, then solution is valid. So, after substitution, \(-8 * -2 + 5\), which is equal to \(21\), thus solution is valid.

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Key Concepts

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

Understanding Algebraic Equations
&quot, unveil the value of \(x\) that is hidden within the numbers and operational signs.
Variable Isolation in Solving Equations
Variable isolation is a pivotal step in solving algebraic equations. It involves manipulating the equation to get the variable by itself on one side of the equal sign. In the given exercise, we start with \( -8x + 5 = 21 \) and aim to isolate \(x\). We do this by first moving the constant term to the other side by adding \(5\) to both sides.

Common Steps for Variable Isolation:

  • Add or subtract constants from both sides to move them away from the variable.
  • Multiply or divide both sides by a number to cancel out coefficients in front of the variable.
  • Simplify both sides of the equation if needed.
Through these steps, we strategically maneuver the equation to eventually express it as \(x = \) some number, which is our goal.
Solution Verification
After finding the value of the variable, it's crucial to ensure that the solution is correct. This is where solution verification comes in鈥攁 step often skipped, but vital for confirming the accuracy of our answer.

For verification, substitute the found value back into the original equation and simplify. If both sides of the equation balance, you've found the correct solution! In our problem, substituting \(x = -2\) into \( -8x + 5 = 21 \) gives us \(16 + 5\), which equals to \(21\), hence verifying that \(-2\) is indeed the correct solution. Remember, if after substitution the equation doesn't balance, we need to revisit our steps to find where we might have gone wrong.

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Most popular questions from this chapter

The circumference \(C\) of a circle is given by \(C=2 \pi r,\) where \(r\) is the radius of the circle. Solve the formula for \(r\), and then calculate the radius of a circle with a circumference of 14.32 feet. Round to the nearest hundredth of a foot.

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