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Chapter 2: Problem 115

Solve the equation \(5 x-6=3 x-8\) (find the value of \(x\) that makes this equation true).

Short Answer

Expert verified
The value of \(x\) that makes the given equation \(5x - 6 = 3x - 8\) true is \(x = -1\).

Step by step solution

01

Isolate x terms on one side

We will start by moving all the terms with \(x\) on the left side of the equation and the constants on the right side. To do that, we will subtract \(3x\) from both sides and add \(6\) to both sides. \[5x - 3x - 6 + 6 = 3x - 3x - 8 + 6\]
02

Simplify both sides of the equation

Now, we will simplify the above equation by performing the operations on both sides: \[2x = -2\]
03

Solve for x

To find the value of \(x\), we will divide both sides of the equation by \(2\): \[\frac{2x}{2} = \frac{-2}{2}\] The final result is: \[x = -1\] So, the value of \(x\) that makes the given equation true is \(-1\).

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are the bedrock of understanding algebra. Think of them as sentences in the language of mathematics, composed of numbers, variables, and operations like addition and subtraction. In the context of our problem, the expression 5x - 6 and 3x - 8 are examples of algebraic expressions.

Variables, like x in our case, are symbols that represent unknown values we aim to find. Numbers that multiply these variables are known as coefficients; here, '5' and '3' are coefficients of x. Constants, like '-6' and '-8', don't have variables attached to them and represent fixed values. By understanding the structure of algebraic expressions, you can manipulate and simplify equations effectively, leading you closer to solving the problem at hand.
The Art of Isolation of Variables
Isolating a variable means rearranging an equation so that the variable we want to solve for is by itself on one side of the equation. This is often a crucial step in solving linear equations. In our example, we aim to isolate the variable x. We achieve this by performing the same operation on both sides of the equation to keep it balanced.

During Step 1 of the solution, we subtracted 3x from both sides effectively moving all x terms to one side. Simultaneously, we added '6' to both sides to move the constants to the opposite side. This left us with a simplified equation having x on one side, a fundamental step towards finding its value. Isolation is an elegant dance of maintaining equality while shifting components to clarify what we are solving.
Simplifying Equations to Reveal Answers
Simplifying equations is akin to reducing clutter to reveal the essence of a problem. It involves combining like terms and performing basic arithmetic. In Step 2 of our solution, we simplified by combining the x terms and the constants after our initial step of isolation. 5x minus 3x gave us 2x, a much simpler expression to work with.

By simplifying, we turned a more complex equation into the straightforward 2x equals negative 2. Doing so made the subsequent operation鈥攄ividing both sides by '2' to solve for x鈥攊ntuitive and effortless. Simplification may not be as flashy as other steps, but it's the meticulous work that sets up success when solving linear equations.

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Most popular questions from this chapter

If 1 U.S. dollar is worth \(1.54\) Canadian dollars, how many U.S. dollars are needed to purchase an item that costs 350 Canadian dollars?

(a) If \(25.0 \mathrm{~cm}^{3}\) of an unknown substance has a mass of \(195 \mathrm{~g}\), what is the density of the substance in grams per cubic centimeter? (b) How many cubic centimeters does \(500.0 \mathrm{~g}\) of the substance occupy? (c) Does this substance sink or float in mercury, which has a density of \(13.6 \mathrm{~g} / \mathrm{mL} ?\)

Without doing any numerical calculations, determine which would have the smallest volume: (a) \(50 \mathrm{~g}\) of water (density \(=1.0 \mathrm{~g} / \mathrm{mL}\) ) (b) \(50 \mathrm{~g}\) of salt water (density \(=2.3 \mathrm{~g} / \mathrm{mL}\) ) (c) \(50 \mathrm{~g}\) of mercury (density \(=13.6 \mathrm{~g} / \mathrm{mL}\) ) (d) \(50 \mathrm{~g}\) of alcohol (density \(=0.89 \mathrm{~g} / \mathrm{mL}\) ) Explain your reasoning.

One liter is equal to \(0.264\) gallon. Suppose you have \(1.000 \times 10^{3} \mathrm{~cm}^{3}\) of water. How many gallons do you have? Use unit analysis to calculate your answer, and show your work. Treat all conversion factors as exact.

How many significant figures are there in each number: (a) \(5.300 \times 10^{-2}\) (b) \(3.2 \times 10^{5}\) (c) \(0.00890 \times 10^{-4}\) (d) \(7.9600000 \times 10^{10}\) (e) \(8.030 \times 10^{21}\)
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