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Chapter 3: Problem 7

Solve these equations. Tell what action you take at each stage. a. \(144 x=12\) b. \(\frac{1}{6} x+2=8\)

Short Answer

Expert verified
a. \(x = \frac{1}{12}\); b. \(x = 36\).

Step by step solution

01

Isolate the Variable for Equation (a)

We start by trying to isolate the variable \(x\) on one side of Equation (a), \(144x = 12\). To do this, we divide both sides of the equation by 144, the coefficient of \(x\). This step will give us the value of \(x\).
02

Solve for x in Equation (a)

Following the division, the left side of the equation becomes \(x\) and the right side becomes \(\frac{12}{144}\), which simplifies to \(\frac{1}{12}\). Thus, the solution to Equation (a) is \(x = \frac{1}{12}\).
03

Simplify Left Side of Equation (b)

For Equation (b), \(\frac{1}{6}x + 2 = 8\), we first aim to eliminate the constant 2 on the left side. Subtract 2 from both sides to simplify and begin isolating \(x\).
04

Isolate the Variable for Equation (b)

After subtraction, the equation becomes \(\frac{1}{6}x = 6\). Now, to isolate \(x\), multiply both sides by 6, which is the reciprocal of the coefficient of \(x\), \(\frac{1}{6}\).
05

Solve for x in Equation (b)

Multiply both sides by 6 to solve for \(x\). Thus, the equation \(\frac{1}{6}x = 6\) becomes \(x = 6 \times 6 = 36\).

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Key Concepts

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

Solving Linear Equations
Solving linear equations involves finding the value of a variable that makes the equation true. These equations are one of the most basic forms of algebra and have a general form like "ax + b = c". In this context:
  • Equation (a): To solve it, we begin by isolating the variable.
  • Equation (b): The aim is also to isolate, and subsequently solve, for the variable.
The process involves using operations that will simplify each side in a balanced way. Remember, whatever action you perform on one side of the equation, you must perform on the other to maintain equality. This principle assures that the equation stays true as you work towards finding the solution.
Isolating Variables
When tackling algebraic equations, isolating variables is a crucial step. It means getting the variable in question, such as 'x', by itself on one side of the equation. Let's see how this is done:
To isolate a variable, you perform operations such as:
  • Division: In Equation (a), dividing by 144 helps isolate 'x' because it cancels the coefficient.
  • Subtraction: In Equation (b), subtracting 2 removes the constant, making the equation simpler.
After isolation, the next step typically involves simplifying the resulting equation, as seen in these examples. This results in a straightforward path to find the exact value of the variable.
Mathematical Operations
Key mathematical operations include addition, subtraction, multiplication, and division. Mastering these is essential for managing equations effectively. Here鈥檚 how they can play out:
  • Division: Used in Equation (a) to eliminate coefficients and make 'x' stand alone.
  • Multiplication: Employed in Equation (b) to reverse division, specifically by using the reciprocal of a fraction like \(\frac{1}{6}\).
These operations allow you to systematically break down and simplify equations until a solution is found. Remember to apply these operations equally to both sides of an equation to maintain balance.

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Most popular questions from this chapter

Solve each equation symbolically using the balancing method. a. \(3+2 x=17\) b. \(0.5 x+2.2=101.0\) c. \(x+307.2=2.1\) d. \(2(2 x+2)=7\) e. \(\frac{4+0.01 x}{6.2}-6.2=0\) (d)

Draw and label a coordinate plane with each axis scaled from \(-10\) to 10 . a. Represent each point named with a dot, and label it using its letter name. \(\begin{array}{lllll}A(3,-2) & B(-8,1.5) & C(9,0) & D(-9.5,-3) & E(7,-4) \\\ F(1,-1) & G(0,-6.5) & H(2.5,3) & I(-6,7.5) & J(-5,-6)\end{array}\) b. List the points in Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. Which points are on the \(x\)-axis? Which points are on the \(y\)-axis? c. Explain how to tell which quadrant a point will be in by looking at the coordinates. Explain how to tell if a point lies on one of the axes.

Suppose a new small-business computer system costs \(\$ 5,400\). Every year its value drops by \(\$ 525\). a. Define variables and write an equation modeling the value of the computer in any given year. b. What is the rate of change, and what does it mean in the context of the problem? c. What is the \(y\)-intercept, and what does it mean in the context of the problem? d. What is the \(x\)-intercept, and what does it mean in the context of the problem?

5\. Give the additive inverse of each number. a. \(\frac{1}{5}\) a b. 17 c. \(-23\) d. \(-x\)

A bicyclist, \(1 \mathrm{mi}(5280 \mathrm{ft})\) away, pedals toward you at a rate of \(600 \mathrm{ft} / \mathrm{min}\) for \(3 \mathrm{~min}\). The bicyclist then pedals at a rate of \(1000 \mathrm{ft} / \mathrm{min}\) for the next \(5 \mathrm{~min}\). a. Describe what you think the plot of (time, distance from you) will look like. (A) b. Graph the data using 1 min intervals for your plot. (a) c. Invent a question about the situation, and use your graph to answer the question.
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