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Chapter 2: Problem 10

Is the given value a solution to the linear equation? $$ -8 x-33=3 x ; \quad x=3 $$

Short Answer

Expert verified
No, \( x = 3 \) is not a solution because \( -57 \neq 9 \).

Step by step solution

01

Substitute x in the Equation

First, we take the given value, \( x = 3 \), and substitute it into the equation \( -8x - 33 = 3x \). This gives us: \( -8(3) - 33 = 3(3) \).
02

Simplify Both Sides of the Equation

Next, simplify both sides of the equation. On the left side, calculate \( -8 \times 3 \) which is \( -24 \), so \( -24 - 33 \). On the right side, calculate \( 3 \times 3 \) which is \( 9 \). Now the equation is \( -24 - 33 = 9 \).
03

Simplify Further

Simplify the left side of the equation further. Calculate \( -24 - 33 \), which equals \( -57 \). The equation now is \( -57 = 9 \).
04

Compare Both Sides of the Equation

Now compare the simplified left side \( -57 \) with the right side \( 9 \). Since \( -57 eq 9 \), the given value \( x = 3 \) is not a solution to the equation.

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Key Concepts

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

Solution Verification
When solving a linear equation, an essential step is to verify whether a proposed value is indeed a solution. Solution verification involves checking if both sides of the equation hold equal after substituting the value. This ensures the correctness of the solution.

To verify if the value \( x = 3 \) is a solution for the equation \( -8x - 33 = 3x \), we replace \( x \) with 3 in the equation.
  • Substitute \( x = 3 \) into the equation: \( -8(3) - 33 = 3(3) \).
  • Solve both sides to check if they equate.
After simplification, if both sides do not match, then \( x = 3 \) is not a solution. In our original exercise, the left side simplified to \( -57 \), whereas the right was \( 9 \), showing that \( x = 3 \) is not a solution.
Substitution Method
The substitution method is a powerful tool in solving equations. It involves replacing variables in an equation with given or previously determined values. This technique is very useful when testing potential solutions because it allows one to directly evaluate the equation without rearranging or balancing it.

In our exercise, we used the substitution method to test \( x = 3 \). Here鈥檚 how it works:
  • Take the equation: \( -8x - 33 = 3x \).
  • Substitute \( x = 3 \) into the equation to get: \( -8(3) - 33 = 3(3) \).
By doing so, you can directly calculate the result of both sides of the equation with the given value. This helps in checking the correctness efficiently.
Simplifying Equations
Simplifying equations is a crucial aspect of algebra that involves reducing an expression to its simplest form. This step makes it easier to compare different terms and helps in identifying solutions.

In our example, simplifying played a key role in verifying the solution:
  • Start by calculating each term after substitution: \( -8(3) = -24 \) and \( 3(3) = 9 \).
  • Then simplify terms to reduce complexity: combine \( -24 - 33 \) on the left side to get \( -57 \).
By simplifying, you can see clearly that the two sides of the equation are not equal (left side: \(-57\), right side: \(9\)). This direct comparison shows that \( x = 3 \) does not satisfy the equation, helping us understand the nature of the solution better.

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Most popular questions from this chapter

Set up a compound inequality for the following and then solve. If two times an angle is between 180 degrees and 270 degrees, then what are the bounds of the original angle?

Set up a compound inequality for the following and then solve. The average temperature in London ranges from \(23^{\circ} \mathrm{C}\) in the summer to \(14^{\circ} \mathrm{C}\) in the winter. Find the equivalent range in degrees Fahrenheit.

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -5<2(x-1)-3(x+2)<5 $$

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Solve and graph. In addition, present the solution set in interval notation. $$ 3(2 x-1)-3(x-2) \leq 2(x+4) $$
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