Problem 3 - Intermediate Algebra

Question

Fill in the blanks. A number that makes an equation true when substituted for the variable is called a _____.

Step-by-Step Solution

Answer

The number is called a solution.

Step 1: Understand the Concept

Begin by identifying what the concept or term is that fits the required blank. In algebraic equations, we're looking for a term describing a number that, when replaced for a variable in an equation, makes the equation hold true.

Step 2: Recall Algebraic Vocabulary

Think about common terminology used in algebra. The term we need is specifically used when discussing equations and solutions to equations.

Step 3: Identify the Term

The correct term is 'solution.' In algebra, a solution is a value or set of values that can be substituted for a variable to make an equation true.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are at the core of algebra. They're statements of equality that involve variables and constants. An algebraic equation asserts that two expressions are equal, such as \(ax + b = c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
Algebraic equations can be used to solve for unknown values in a variety of contexts. They are a critical tool for everything from basic arithmetic problems to complex calculations in science and engineering.
To solve an algebraic equation, you'll typically perform a series of operations to isolate the variable. This involves:

• Adding or subtracting terms on both sides
• Multiplying or dividing both sides by the same number
• Reversing the operations to find the value of the variable

Understanding algebraic equations is essential for developing a strong foundation in mathematical problem-solving.

Substitute Variable Explained

Substitution is a key concept in algebra. It involves replacing a variable in an equation with a specific value. This process helps us verify whether a particular value is a solution to the equation.
For example, consider the equation \(3x + 4 = 10\). If you suspect that \(x = 2\) is a solution, you can substitute \(2\) in place of \(x\) in the equation:\[3(2) + 4 = 10\]
This substitution helps you check if the equation becomes true. If the equation is balanced, that means \(x = 2\) is indeed a solution. If not, you'll have to try another value.
Substitution is not just about finding solutions. It is also a vital tool in developing problem-solving skills. It helps us make informed decisions about which values satisfy given conditions or equations.

Defining a True Equation

A true equation is one where both sides are equal after substituting the variable with the correct value or values. An equation like \(5x + 3 = 18\) is true only when a specific substitution satisfies it completely.
Finding a true equation requires:

• Identifying the relevant substitutes for variables
• Ensuring calculations for both sides of the equation are correct
• Guaranteeing that the equation balances, meaning that its left and right sides result in the same value

For example, if we take the equation \(2x + 5 = 11\), substituting \(x = 3\) gives:\[2(3) + 5 = 11\]
Here, the equation is true because both sides equal 11 with the substitution. Understanding what makes an equation true helps in recognizing which values genuinely represent solutions. This understanding can expand to more complex mathematical relationships, reinforcing accuracy and reliability in problem-solving.