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Chapter 2: Problem 33

Solve each equation, and check the solution. \(6(3-x)=-6 x+18\)

Short Answer

Expert verified
The equation is an identity and has infinitely many solutions.

Step by step solution

01

Distribute the 6 on the left side

First, use distribution on the left side of the equation to eliminate the parentheses. Multiply 6 by each term inside the parentheses: \[ 6(3 - x) = 6 \times 3 - 6 \times x = 18 - 6x \] So, the equation becomes: \[ 18 - 6x = -6x + 18 \]
02

Move variable terms to one side

Next, add \(6x\) to both sides of the equation to isolate the variable terms on one side. \[ 18 - 6x + 6x = -6x + 6x + 18 \] Now, simplify: \[ 18 = 18 \]
03

Evaluate the solution

The remaining equation \[ 18 = 18 \] is always true, which means that any value of \(x\) will satisfy the original equation. Therefore, the original equation is an identity and has infinitely many solutions.

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Key Concepts

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

distribution
Distribution is a key mathematical property. It helps simplify equations, eliminating parentheses to make them easier to solve. In our exercise, we distributed the value 6 over the terms inside the parentheses (3 - x). Here's how we did it:
First, take the value outside the parentheses (6) and multiply it by each term inside:
  • 6 times 3 = 18
  • 6 times -x = -6x
After distribution, our equation looked like this:
18 - 6x = -6x + 18.
isolating variables
Isolating variables means rearranging an equation so that the variable (like x) stands alone on one side. This helps us find its value. In our example, after distribution, we had:
18 - 6x = -6x + 18
To isolate the variable, we need all x terms on one side. We added 6x to both sides:
18 - 6x + 6x = -6x + 6x + 18
By simplifying, it becomes:
18 = 18.
Here, adding 6x cancelled out the x terms, showing this equation has a special property.
identity equation
An identity equation is an equation that is always true, no matter what value is chosen for the variable. In our example, isolating the variable simplified our equation to:
18 = 18
This is an obvious truth. No matter what value we pick for x, both sides of the equation will always be equal. Hence, it's called an identity equation. This special property signifies that our original equation was an identity.
infinitely many solutions
When an equation is true for any value of the variable, it has infinitely many solutions. In our solved equation 18 = 18, there are no particular restrictions on x. We didn't end up with a specific value for x at the end.
This means that x can be any real number, and it will still make the equation true. These types of equations are unique because they are satisfied by infinitely many values. In every instance, the left side and right side are equal, confirming that the solution sets are all real numbers.

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Most popular questions from this chapter

The world's largest taco contained approximately \(1 \mathrm{~kg}\) of onion for every \(6.6 \mathrm{~kg}\) of grilled steak. The total weight of these two ingredients was \(617.6 \mathrm{~kg}\). To the nearest tenth of a kilogram, how many kilograms of each ingredient were used to make the taco? (Data from Guinness World Records.

If 6 is subtracted from the third of three consecutive odd integers and the result is multiplied by \(2,\) the answer is 23 less than the sum of the first and twice the second of the integers. Find the integers.

Two apartments have numbers that are consecutive integers. The sum of the numbers is 59\. What are the two apartment numbers?

Solve each inequality. Graph the solution set, and write it using interval notation. \(5<1-6 m<12\)

Find the measure of an angle such that the sum of the measures of its complement and its supplement is \(160^{\circ} .\)
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