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Chapter 2: Problem 15

Solve each equation, and check your solution. If applicable, tell whether the equation is an identity or a contradiction. See Examples \(2,3,\) and \(6 .\) $$ 7 x-5 x+15=x+8 $$

Short Answer

Expert verified
The solution is x = -7. The equation is a conditional equation.

Step by step solution

01

- Combine like terms

Combine like terms on the left side of the equation. The terms are 7x and -5x:7x - 5x + 15 = x + 8This simplifies to:2x + 15 = x + 8
02

- Isolate the variable

Subtract x from both sides to isolate the variable terms on one side of the equation:2x - x + 15 = x - x + 8This simplifies to:x + 15 = 8
03

- Solve for x

Subtract 15 from both sides to solve for x:x + 15 - 15 = 8 - 15This simplifies to:x = -7
04

- Check the solution

Substitute the value of x back into the original equation to check the solution:7(-7) - 5(-7) + 15 = -7 + 8Simplify each side:-49 + 35 + 15 = 11 = 1Since both sides of the equation are equal, the solution x = -7 is correct.
05

- Classify the equation

Since the equation has a solution and is not true for all values of x, it is neither an identity nor a contradiction. It is a conditional equation.

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Key Concepts

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

Combining like terms
When solving linear equations, one of the first steps is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, the terms with the variable x are 7x and -5x. We combine these by adding their coefficients:
7x - 5x = 2x.
This reduces our equation from 7x - 5x + 15 = x + 8 to 2x + 15 = x + 8.
Combining like terms makes the equation simpler and easier to solve. Always look for like terms first before moving to other steps.
Isolating the variable
After combining like terms, the next step in solving a linear equation is isolating the variable. Isolating the variable means getting the variable on one side of the equation by itself.
We do this by performing the same mathematical operation to both sides of the equation.
In our example, we have 2x + 15 = x + 8. Subtract x from both sides to start: 2x - x + 15 = x - x + 8
This simplifies to x + 15 = 8.
Now, subtract 15 from both sides: x + 15 - 15 = 8 - 15
This gives us x = -7.
By isolating the variable, we determine the specific value that solves the equation.
Checking solutions
Once we find a solution for the variable, it is essential to check if this solution is correct by substituting it back into the original equation. Let鈥檚 check our solution of x = -7 in the original equation 7x - 5x + 15 = x + 8.
Substitute -7 for x:
7(-7) - 5(-7) + 15 = -7 + 8.
Simplify each side:
-49 + 35 + 15 = -7 + 8.
The left side simplifies to 1, and the right side simplifies to 1 as well. Since both sides are equal, our solution x = -7 is correct.
Checking our solutions ensures that we have not made any errors in our calculations.
Identities and contradictions in equations
In some cases, an equation can be an identity or a contradiction. An identity is true for all values of the variable. For example, x = x. On the other hand, a contradiction is never true regardless of the variable's value, such as x = x + 1.
For our example, 7x - 5x + 15 = x + 8 simplifies to x = -7, which is only true for x = -7 and not for other values. Therefore, this is considered a conditional equation.
Understanding whether an equation is an identity, a contradiction, or conditional helps categorize the nature of the solutions we find.

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Most popular questions from this chapter

As mentioned in the chapter introduction, 114.9 million U.S. households owned at least one TV set in 2009. (Source: Nielsen Media Research.) Use this information to work Exercises 43-46. Round answers to the nearest percent in Exercises \(43-44,\) and to the nearest tenth million in Exercises \(45-46 .\) About 62.0 million U.S. households owned 3 or more TV sets in \(2009 .\) What percent of those owning at least one TV set was this?

Solve each problem about percent increase or percent decrease. Between 2000 and 2007 , the estimated population of Anchorage, Alaska, grew from \(320,391\) to \(362,340 .\) What was the percent increase to the nearest tenth?

In each of the following, (a) translate as an expression and (b) translate as an equation or inequality. Use x to represent the number. (a) 8 less than a number (b) 8 is less than a number.

When \(\frac{2}{3}\) of a number is subtracted from \(14,\) the result is \(10 .\) Find the number.

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) Sondra Braeseker bought a new car and agreed to pay it off in 36 monthly payments. The total finance charge was \(\$ 700 .\) Find the unearned interest if she paid the loan off 4 payments ahead of schedule.
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