/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Solve the given equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 1

Solve the given equation. $$ 3 x=12 $$

Short Answer

Expert verified
The short answer to the equation \(3x = 12\) is \(x = 4\).

Step by step solution

01

Identify the coefficient of x

In the equation \(3x = 12\), the coefficient of x is 3. This means that x is being multiplied by 3.
02

Divide both sides of the equation by the coefficient of x

To isolate x, we need to divide both sides of the equation by the coefficient (3). This will allow us to get x by itself on one side of the equation. \(3x = 12\) \(\frac{3x}{3} = \frac{12}{3}\)
03

Simplify the equation

Now that we've divided both sides of the equation by 3, we can simplify. \(x = 4\)
04

State the solution

The solution to the equation \(3x = 12\) is \(x = 4\).

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Key Concepts

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

Isolate the Variable
In algebra, isolating the variable is a fundamental step in solving equations. It involves rearranging the equation so that the variable you are solving for is on one side of the equation by itself. This allows you to clearly see the value of the variable.

For a simple equation like \(3x = 12\), the goal is to get \(x\) on one side of the equal sign without any numbers attached to it. You can think of the equation as a balanced scale: whatever you do to one side, you must do to the other to keep it balanced. In this case, since \(x\) is multiplied by 3, you can balance the scale by dividing both sides by 3, effectively reversing the multiplication. This leaves you with \(x = 4\), isolating the variable completely.
Coefficient
The term 'coefficient' in algebra refers to the number that is multiplied by the variable in an algebraic expression. It is an essential component of linear equations. Understanding coefficients is key to manipulating equations effectively.

In the example \(3x = 12\), the number 3 is the coefficient of \(x\). It indicates how much \(x\) is scaled by in the equation. When solving equations, we often modify the coefficient to manipulate the equation, such as dividing by the coefficient to cancel it out. This action is designed to simplify the equation and bring us one step closer to finding the value of the variable.
Algebraic Manipulations
Algebraic manipulations are the various operations you can apply to an equation to solve for an unknown variable. These operations, such as adding, subtracting, multiplying, and dividing, should always maintain the balance of the equation. Manipulations must follow the properties of equality and can also include factoring, distributing, and combining like terms.

In solving \(3x = 12\), we used the manipulation of division to both sides to simplify the equation. This isolated the variable effectively. It's also important when doing algebraic manipulations to be cautious of the order of operations, sometimes known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to make sure that we perform the operations in the correct sequence for accurate results.

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Most popular questions from this chapter

The distribution of income in a certain city can be described by the mathematical model \(y=\left(2.8 \cdot 10^{11}\right)(x)^{-1.5}\), where \(y\) is the number of families with an income of \(x\) or more dollars. a. How many families in this city have an income of \(\$ 20,000\) or more? b. How many families have an income of \(\$ 40,000\) or more? c. How many families have an income of \(\$ 100,000\) or more?

Perform the indicated operations and simplify. \(\frac{\frac{1}{x^{3}}-\frac{1}{y^{3}}}{\frac{1}{x}-\frac{1}{y}}\)

Solve the equation. $$ \frac{3 x}{x+1}+\frac{2}{x}+5=\frac{3}{x^{2}+x} $$

The average speed of a vehicle in miles per hour on a stretch of route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the expression $$ 20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. Over what interval of time is the average speed of a vehicle less than or equal to \(35 \mathrm{mph}\) ?

Evaluate the expression. $$ |\pi-1|+2 $$
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