Algebra Chapter on Isolating x

Questions and Answers

What are the steps for solving $x$ when $x$ is in the numerator?

Divide the numbers 2. Multiply all terms by the same value to eliminate fraction denominators 3. Simplify 4. Write $x = $

Solve $\frac{14}{7} = \frac{x}{12}$

$x = 24$

What are the steps for solving $x$ when $x$ is in the denominator?

Divide the numbers 2. Multiply all terms by the same value to eliminate fraction denominators 3. Simplify 4. Multiply all terms by the same value to eliminate fraction denominators 5. Simplify 6. Divide both sides of the equation by the same term 7. Write $x = $

Solve $\frac{12}{x} = \frac{4}{6}$

$x = 18$ Signup and view all the answers

What does it mean to isolate the variable?

Isolating a variable means rearranging an algebraic equation so that a different variable is on its own. Signup and view all the answers

What is the basic technique to isolate a variable?

The basic technique is to do something to both sides of the equation, such as add, subtract, multiply, or divide by the same number. Signup and view all the answers

What is transposition?

Transposition is a method to isolate the variable to one side of the equation and everything else to the other side. Signup and view all the answers

Algebraic equations can be solved using the Law of Equations.

True Signup and view all the answers

How do you isolate a variable in the denominator?

To isolate a variable in the denominator, you cross-multiply the equation and collect like terms. Signup and view all the answers

What are the steps to solve the example $\frac{1}{3}x = 8$?

<ol> <li>Cross multiply: $3x \times 8 = 1$ 2. $24x = 1$ 3. Divide both sides by 24, so, $x = \frac{1}{24}$</li> </ol> Signup and view all the answers

What are the steps to solve the example $\frac{3}{x} = 3$?

<ol> <li>Cross multiply the equation: $3x = 3$ 2. Divide both sides by 3 to isolate $x$ 3. $x = 1$</li> </ol> Signup and view all the answers

What are the rules for solving basic equations with addition?

Subtract the digit from both sides to isolate the variable. Signup and view all the answers

What are the rules for solving basic equations with subtraction?

You add $x$ on both sides and then continue to isolate it. Signup and view all the answers

What are the two rules to remember when multiplying negative numbers?

<ol> <li>Two negative values give a positive value. 2. If there is one negative value, the answer is negative.</li> </ol> Signup and view all the answers

Study Notes

Solving for x in Numerators

Steps to solve for 𝑥 when it is in the numerator include dividing numbers, multiplying all terms by a common value to eliminate fraction denominators, simplifying, and then writing 𝑥 = result.

Example: Solving 14/7 = 𝑥/12

Initial division gives 2 = 𝑥/12.
Multiply all terms by 12 to clear the denominator: 24 = 𝑥.

Solving for x in Denominators

For 𝑥 in the denominator, similarly divide, then clear denominators through multiplication, simplify, and ultimately isolate 𝑥.

Example: Solving 12/𝑥 = 4/6

Simplification leads to 12 = 2𝑥/3, followed by eliminating fractions and solving for 𝑥 = 18.

Understanding Variable Isolation

Isolating a variable entails rearranging an algebraic equation to have the variable of interest on one side and all other terms on the opposite side.

Basic Technique for Isolation

To isolate a variable, perform the same operation on both sides of the equation multiple times as necessary until the variable is alone.

Definition of Transposition

Transposition is a strategy for isolating a variable, letting one side of the equation contain just that variable.

Law of Equations

This law asserts that any operation applied to one side of an equation must also be performed on the opposite side to maintain equality.

Isolating Variables in the Denominator

To isolate variables in the denominator, cross-multiply and then collect like terms.

Example: 1/3𝑥 = 8

Cross multiplication results in 24𝑥 = 1, leading to 𝑥 = 1/24 after division.

Example: 3/𝑥 = 3

Cross multiplying yields 3𝑥 = 3, resulting in 𝑥 = 1 after division by 3.

Practice Questions for Isolation

Several practice equations are provided for students to isolate 𝑥, such as 8/𝑥 + 1 = 4/3 and 4 - 3𝑥 = 40.

Addition Rules in Equation Solving

For equations involving addition, subtract the constant from both sides to isolate 𝑥. This applies to both forms of addition equations.

Subtraction Rules in Equation Solving

Equations with subtraction involve adding the variable to both sides first, then subtracting the remaining term from both sides to isolate 𝑥.

Rules for Multiplying Negative Numbers

Multiplying two negative numbers results in a positive value, while multiplying one negative number yields a negative result.

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Description

This quiz focuses on solving for the variable x when it appears in the numerator and denominator. Students will learn the steps to isolate x, manipulations needed to clear denominators, and practical examples for better understanding. Prepare to apply algebraic techniques to different scenarios involving x!