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Chapter 1: Problem 14

Determine whether each of the following is an expression or an equation. \(-10 x+12-4 x+3=0\)

Short Answer

Expert verified
Equation.

Step by step solution

01

- Identify Elements

Look at the given mathematical statement \(-10x + 12 - 4x + 3 = 0\). Identify the components: numbers, variables, operators, and the equal sign.
02

- Determine Presence of Equality Sign

Check if there is an equal sign \( = \). This will help distinguish between an equation and an expression. An equation will have an equal sign dividing two expressions, while an expression will not.
03

- Define Expression and Equation

An expression is a mathematical phrase that can contain numbers, variables, and operators but does not contain an equal sign. An equation is a mathematical statement that asserts the equality of two expressions, containing an equal sign.
04

- Apply Definition

Since it contains the equal sign \( = \), the given statement \(-10x + 12 - 4x + 3 = 0\) is classified as an equation.

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Key Concepts

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

mathematical expressions
A mathematical expression is like a phrase in arithmetic. It can include numbers, variables (which are letters representing numbers), and operators (like +, 鈥�, *, and /). However, it does not have an equal sign. This means an expression just shows a combination of terms, but it doesn't assert that two things are equal. For example, **10 + 5 - 3x** is an expression. It's important to understand that expressions can be simplified or evaluated. Simplifying an expression involves combining like terms, while evaluating means substituting the values into the variables and performing the arithmetic operations.
equations
An equation is like a complete sentence in mathematics. It has two sides separated by an equal sign ( = ). Each side is an expression. An equation asserts that the two expressions on either side of the equal sign are equal. For example, **10 - x = 5** is an equation because it claims that subtracting x from 10 results in 5. To solve an equation, you need to find the value of the variable that makes the equation true. This process involves various algebraic techniques, like isolating the variable or using inverse operations.
variables and constants
In algebra, variables are symbols (often letters like x, y, or z) that stand for unknown numbers. They can take on different values. Constants, on the other hand, are specific, unchanging numbers. For example, in the expression **3x + 5**, **x** is a variable, while 3 and 5 are constants. Variables are essential because they allow us to form general mathematical statements and equations. Constants give us fixed points of reference within those expressions or equations. Understanding the difference between variables and constants is crucial for simplifying expressions and solving equations.

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Most popular questions from this chapter

To earn a \(\mathrm{B}\) in an algebra course requires an average of at least 80 on five tests. A student has scores of \(75,91,82,\) and \(74 .\) What possible scores on the fifth test would guarantee this student a \(\mathrm{B}\) in the class?

Solve each equation or inequality. Graph the solution set. $$ |2 x-6| \leq 6 $$

Complete each statement. The following key terms may be used once, more than once, or not at all. $$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing \end{array}$$ A linear equation with one solution in its ____________, such as \(2 x+5=9,\) is \(\mathrm{a}(\mathrm{n})\) ___________.

To solve a linear equation with decimals, we usually begin by multiplying by a power of 10 so that all coefficients are integers. What is the least power of 10 that will accomplish this goal in each equation? (a) \(0.05 x+0.12(x+5000)=940\) (b) \(0.006(x+2)=0.007 x+0.009\)

Complete the six problem-solving steps to solve each problem. In 2015 , the corporations securing the most U.S. patents were IBM and Samsung. Together, the two corporations secured a total of 12,368 patents, with Samsung receiving 2250 fewer patents than IBM. How many patents did each corporation secure? (Data from U.S. Patent and Trademark Office.) Step 1 Read the problem carefully. We are asked to find ______ . 2 Assign a variable. Let \(x=\) the number of patents that IBM secured. Step Then \(x-2250=\) the number of ______ . Step 3 Write an equation. ____\(+ _____ \ldots=12,368\) Step 4 Solve the equation. \(x=\) ______ . \(\begin{array}{llll}\text { Step } 5 & \text { State the answer. } & \text { IBM secured } & \text { patents. Samsung secured } & \text { patents. }\end{array}\) Step 6 Check. The number of Samsung patents was _____ fewer than the number of _____ The total number of patents was \(7309+\) _____ \(= ______\).
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