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Chapter 2: Problem 104

For exercises 103-106, solve the equation. Use a calculator to do the arithmetic. $$ 901 z+302=2(492 z+3)-83 z $$

Short Answer

Expert verified
No solution, the equation \( 302 = 6 \) is a contradiction.

Step by step solution

01

- Distribute the terms on the right side

Distribute the 2 on the right side of the equation: \( 2(492z + 3) = 2 \times 492z + 2 \times 3 = 984z + 6 \). So the equation becomes: \( 901z + 302 = 984z + 6 - 83z \).
02

- Combine like terms on the right side

Combine the terms involving z on the right side: \( 984z - 83z + 6 = 901z + 6 \). So the equation is now: \( 901z + 302 = 901z + 6 \).
03

- Subtract 901z from both sides

Subtract 901z from both sides of the equation: \( 901z + 302 - 901z = 901z + 6 - 901z \). This simplifies to: \( 302 = 6 \).
04

- Analyze the resulting equation

The simplified equation is \( 302 = 6 \), which is a contradiction. This means there are no values of \( z \) that satisfy the original equation.

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Key Concepts

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

Distributive Property
When solving equations, the distributive property is a crucial tool. It allows us to multiply a single term across terms within parentheses. Here's how it works: if you have an expression like 2(3 + 4), you can distribute the 2 to both the 3 and the 4. This gives you:
2(3 + 4) = 2 * 3 + 2 * 4 = 6 + 8 = 14.
In the context of our exercise, we used the distributive property to handle the right side of the equation:
2(492z + 3) = 2 * 492z + 2 * 3 = 984z + 6.
By distributing the 2 across both 492z and 3, we simplify the equation and make the next steps possible. Remember, distributing means multiplying the term outside the parentheses by each term inside.
Combining Like Terms
After distributing, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In equation problems, combining like terms helps to simplify the equation. For example, if you're given:
7x + 3x - 2 = 0,
you can combine the 7x and 3x since they both contain the variable x. This gives you:
(7 + 3)x - 2 = 0,
which simplifies to:
10x - 2 = 0.
In our given problem, after distribution, we combined like terms on the right side:
984z - 83z + 6 = 901z + 6.
This simplification is important for making the equation easier to solve.
Contradiction in Equations
Sometimes, while solving equations, we end up with an equation that is always false, regardless of the variable's value. This is called a contradiction. For example, if you simplify an equation and end up with:
0 = 5,
this statement is clearly false, as zero is never equal to five. In such a case, we say the original equation has no solution. In our step-by-step solution, we found a contradiction:
302 = 6.
This equation is also false and indicates that the original equation has no solutions. Contradictions often arise when combining and simplifying terms, and they signal the impossibility of finding a value that satisfies the equation.
No Solution
When an equation leads to a contradiction, we determine that it has no solutions. This means there are no values for the variable that make the equation true. In our exercise, after simplifying and combining terms, we got:
302 = 6.
Since this statement is false, we identified that the equation has no solution. Recognizing no solution is crucial as it tells us our original equation is inconsistent. When solving equations, always verify if your simplified results make sense. If you get an inherently false statement, it's a clear indicator of no solution.

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Most popular questions from this chapter

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ 20>-2 u+40 $$

The cost of one double orchid lei from The Hawaiian Lei Company is \(\$ 25.08\). Find the total cost for an order of three double orchid leis sent to the mainland, including the FedEx shipping charge and the shipping surcharge. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number } \\ \text { of leis } \end{array} & \begin{array}{c} \text { FedEx } \\ \text { shipping charge } \end{array} & \text { Surcharge } \\ \hline 1 & \$ 17.92 & \$ 8 \\ 2 & \$ 21.28 & \$ 8 \\ 3 & \$ 25.52 & \$ 8 \\ 4 & \$ 26.88 & \$ 8 \\ \hline \end{array} $$ $$ \text { Source: www.hawaiianleicompany.com } $$

For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ 3(4 x-1) \leq 9(x-3) $$

For exercises \(65-72\), assign a variable, and write an inequality that represents the constraint. To qualify for a group discount, a group must have at least 14 people.

For exercises 1-8, use a number line graph to represent the inequality. $$ -7
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