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Chapter 7: Problem 20

Fill in the blanks. In the term \(y\), the _____ is understood to be 1.

Short Answer

Expert verified
In the term \(y\), the 'coefficient' is understood to be 1.

Step by step solution

01

Understand the concept of coefficient

In algebra, a coefficient is a numero that multiplies a term. For example, in 3x, 3 is the coefficient of x.
02

Apply the coefficient rule

When a variable appears without any specific coefficient, the coefficient is understood to be 1. This applies to any variable on its own.
03

Fill in the blank

So, in the term y, the missing term in the blank, which is the understood coefficient, is 1. We fill in the blank with 'coefficient'

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Key Concepts

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

Coefficient
In algebra, a **coefficient** is the number that stands in front of a variable to indicate how many times the variable is multiplied. It is like a multiplying partner for the variable. For instance:
  • In the term \(3x\), the number 3 is the coefficient, meaning 3 times the variable \(x\).
  • If you see \(7y^2\), then 7 is the coefficient, showing 7 times \(y\) squared.
Understanding coefficients is important because they help in figuring out how variables contribute to the total value in expressions or equations. Sometimes, a variable appears alone, like \(y\), with no number in front. In this case, a silent rule tells us that the coefficient is 1. So, \(y\) can be thought of as \(1y\). This rule holds true for any solitary variable, clarifying calculations and simplifying expressions.
Variable
A **variable** in algebra is a symbol, usually a letter, that represents an unknown or changeable quantity. Variables are essential in forming equations and expressions, making them flexible tools for solving problems. Commonly used letters include \(x\), \(y\), and \(z\). Here are some key points about variables:
  • Variables can take on different values, which means they can change depending on the equation.
  • In expressions, they are often paired with numbers or coefficients to show that they multiply together.
  • Variables help model real-world scenarios mathematically, such as calculating distances, speeds, and many other phenomena.
In the simple term \(x + 5\), \(x\) is the variable, showing an unknown number added to 5. By working with variables, algebra allows for the creation of generalized formulas and the solving of complex problems.
Terms in Algebra
**Terms in algebra** are the building blocks of expressions and equations. Each term can consist of numbers, variables, or both. They are connected by addition or subtraction in an algebraic expression. In general, a term can be:
  • A number only, known as a constant, like \(7\) or \(-3\).
  • A variable, such as \(x\) or \(y\).
  • A combination of a coefficient and a variable, like \(4x\) or \(-2y^2\).
Terms are separated in an expression by plus \((+)\) or minus \((-)(\) signs. For example, in the expression \(3x + 2y - 5\), there are three distinct terms: \(3x\), \(2y\), and \(-5\). Knowing how to identify and work with terms helps in simplifying expressions, solving equations, and understanding how different elements of algebra interact.

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Most popular questions from this chapter

Solve each system using any method. $$\left\\{\begin{array}{l}\frac{1}{2} x-\frac{1}{3} y=-2 \\\x+2 y=4\end{array}\right.$$

Graph the solution. $$y<2-3 x$$

Graph the solution. $$\left\\{\begin{array}{l}\frac{x}{3}-\frac{y}{2}<-3 \\\\\frac{x}{3}+\frac{y}{2}>-1\end{array}\right.$$

Solve each system using any method. $$\left\\{\begin{array}{l}3 x+y=5 \\\3 x-4 y=10\end{array}\right.$$

Use two equations in two variables to solve each application. A man drives for a while at 45 mph. Realizing that he is running late, he increases his speed to \(60 \mathrm{mph}\) and completes his 405 -mile trip in 8 hours. How long does he drive at \(45 \mathrm{mph} ?\)
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