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Chapter 6: Problem 63

Explain what it means to solve a formula for a variable.

Short Answer

Expert verified
Solving a formula for a variable means applying algebraic principles to find the value of the variable that makes the equation true, usually by isolating the variable on one side of the equation, such as in the equation \(2x + 5 = 13\), where \(x = 4\).

Step by step solution

01

Define the Concept

Solving a formula for a variable, also known as solving an algebraic equation, involves finding the value of the variable that makes the equation true. This usually means isolating the variable on one side of the equation while moving all other components of the equation to the opposite side.
02

Explain the Principle of Isolation

In order to isolate the variable, one uses basic algebraic principles such as addition, subtraction, multiplication and division. They are applied to both sides of the equation to maintain the equality. For example, if we have the equation \(2x + 5 = 13\), we would like to solve this for the variable \(x\), meaning we want to isolate \(x\). This is achieved by applying an operation to both sides of the equation.
03

Provide an Example

Continuing with the previous example \(2x + 5 = 13\), we can start by subtracting 5 from both sides of the equation. This gives us \(2x = 13 - 5\) or \(2x = 8\). Now to get \(x\), we divide both sides of the equation by 2, resulting in \(x = 8 / 2\) or \(x = 4\). Thus \(x=4\) is the solution to the formula.

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Key Concepts

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

Isolate the Variable
When we talk about isolating the variable in algebra, we are referring to the process of rearranging an equation so that the variable we're solving for is by itself on one side of the equation. The goal is to find the variable's value that makes the equation true.

For instance, consider the equation \(3y - 6 = 9\). To isolate \(y\), we would first add 6 to both sides to get \(3y = 15\). Following that, we divide both sides by 3, and \(y\) is now alone on one side: \(y = 5\). The variable \(y\) is referred to as being isolated when it stands by itself, with no other terms or coefficients attached to it. This is a fundamental step in solving algebraic equations.
Basic Algebraic Principles
Understanding some basic algebraic principles is crucial to solving equations effectively. These principles, such as the commutative property (e.g., \(a + b = b + a\)), associative property (e.g., \(a + (b + c) = (a + b) + c\)), and distributive property (e.g., \(a(b + c) = ab + ac\)), serve as the foundation of algebraic manipulation.

Another critical principle is doing the same operation to both sides of an equation, which maintains the balance or equality of the equation. For example, if you add 3 to one side of an equation, you must add 3 to the other side as well. These principles guide us in transforming equations to isolate variables and solve for them.
Equality in Equations
The concept of equality is at the heart of every algebraic equation. An equation represents two expressions that are equal in value, symbolized by the 鈥榎=(\=)鈥� sign. It鈥檚 like a see-saw; whatever changes you make on one side must be made on the other to keep it balanced.

If you multiply one side by a number, you must do the same to the other. For example, if \(x/2 = 7\), and we multiply both sides by 2, we maintain equality and find that \(x = 14\). Remember, preserving this balance is essential when performing algebraic operations to solve the equation.
Algebraic Operations
Algebraic operations, including addition, subtraction, multiplication, and division, are the tools we use to manipulate equations. They allow us to move terms around in an equation and to isolate the variable.

Let鈥檚 take the equation \(2x + 3 = 7\) as an example. We can subtract 3 from both sides (\

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Most popular questions from this chapter

Solve the quadratic equations in Exercises 37-52 by factoring. \(x^{2}-2 x-15=0\)

Solve each equation by the method of your choice. When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.

Solve each equation by the method of your choice. When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Explain how to solve a quadratic equation using the quadratic formula. Use the equation \(x^{2}+6 x+8=0\) in your explanation.

In an inequality such as \(5 x+4<8 x-5\), I can avoid division by a negative number depending on which side \(I\) collect the variable terms and on which side I collect the constant terms.
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