Solving Linear Equations in Algebra

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Questions and Answers

What is the goal of solving a linear equation?

To eliminate any constants from the equation

To isolate the variable on one side of the equation (correct)

To eliminate any coefficients from the equation

To rewrite the equation in a different form

What is the final step in solving a linear equation?

Add or subtract the same value to both sides of the equation

Read and understand the problem statement

Simplify the equation to get the final solution (correct)

Multiply or divide both sides of the equation by a coefficient

What property of equation manipulation allows you to add or subtract the same value to both sides of an equation?

Addition and subtraction property (correct)

Equivalent equations property

Multiplication and division property

Linear equation property

What type of linear equation word problem involves finding the cost of one unit of a product?

Cost problem Signup and view all the answers

What is the definition of equivalent equations?

Equations that have the same solution Signup and view all the answers

What is the first step in solving a linear equation word problem?

Read and understand the problem statement Signup and view all the answers

What is the purpose of equation manipulation?

To rewrite an equation in a different form Signup and view all the answers

What is the result of multiplying both sides of an equation by a non-zero value?

The equation becomes equivalent Signup and view all the answers

What is the main reason for combining like terms on each side of the equation when solving a linear equation?

To ensure the equation is balanced Signup and view all the answers

What should you do if you get a contradiction when solving a linear equation?

There is no solution to the equation Signup and view all the answers

Why is it important to check the solution to a linear equation?

To ensure the solution is correct Signup and view all the answers

What is the main difference between a linear equation with no solution and a linear equation with infinite solutions?

The truth value of the equation Signup and view all the answers

When solving a linear equation, what operation should you perform to get rid of a constant term?

Add the same value to both sides Signup and view all the answers

What is the purpose of writing an equation based on the information given in a word problem?

To identify the unknown variable Signup and view all the answers

Study Notes

Solving Linear Equations

A linear equation in one variable is an equation of the form ax + b = c, where a, b, and c are constants, and x is the variable.
To solve a linear equation, the goal is to isolate the variable x on one side of the equation.
Steps to solve a linear equation:
1. Add or subtract the same value to both sides of the equation to eliminate any constants.
2. Multiply or divide both sides of the equation by a coefficient (a value multiplied by the variable) to eliminate any coefficients.
3. Simplify the equation to get the final solution.

Linear Equation Word Problems

Linear equation word problems involve using linear equations to model real-world scenarios.
Steps to solve a linear equation word problem:
1. Read and understand the problem statement.
2. Identify the variable and the constants involved.
3. Write a linear equation based on the information given.
4. Solve the equation using the steps mentioned earlier.
5. Interpret the solution in the context of the problem.

Examples of linear equation word problems:

Cost problem: If it costs $x to buy x units of a product, and the total cost is $120 when buying 8 units, how much does one unit cost?
Distance problem: If a car travels at a speed of x miles per hour, and it travels 240 miles in 4 hours, how fast was the car traveling?

Equation Manipulation

Equation manipulation involves rearranging an equation to get it into a different form.
Addition and subtraction property: The same value can be added or subtracted to both sides of an equation without changing its solution.
Multiplication and division property: Both sides of an equation can be multiplied or divided by the same non-zero value without changing its solution.
Equivalent equations: Two equations that have the same solution are called equivalent equations.

Examples of equation manipulation:

Simplifying an equation: 2x + 3 = 7 can be simplified to 2x = 4 by subtracting 3 from both sides.
Rearranging an equation: x - 2 = 5 can be rearranged to x = 7 by adding 2 to both sides.

Solving Linear Equations

A linear equation in one variable is of the form ax + b = c, where a, b, and c are constants, and x is the variable.
The goal is to isolate the variable x on one side of the equation.
Steps to solve a linear equation:
- Add or subtract the same value to both sides to eliminate constants.
- Multiply or divide both sides by a coefficient to eliminate coefficients.
- Simplify the equation to get the final solution.

Linear Equation Word Problems

Linear equation word problems involve modeling real-world scenarios with linear equations.
Steps to solve a linear equation word problem:
- Read and understand the problem statement.
- Identify the variable and constants involved.
- Write a linear equation based on the given information.
- Solve the equation using the steps mentioned earlier.
- Interpret the solution in the context of the problem.

Equation Manipulation

Equation manipulation involves rearranging an equation to get it into a different form.
The addition and subtraction property allows adding or subtracting the same value to both sides of an equation.
The multiplication and division property allows multiplying or dividing both sides of an equation by the same non-zero value.
Equivalent equations have the same solution.
Examples of equation manipulation:
- Simplifying an equation: 2x + 3 = 7 can be simplified to 2x = 4.
- Rearranging an equation: x - 2 = 5 can be rearranged to x = 7.

Solving Linear Equations

Definition

A linear equation in one variable is an equation in the form ax + b = 0, where a and b are constants, and x is the variable.

Steps to Solve a Linear Equation

Simplify the equation by combining like terms and removing parentheses.
Isolate the variable by adding or subtracting the same value to both sides to get rid of the constant term, and then dividing both sides by the coefficient of the variable.
Check the solution by plugging it back into the original equation to verify that it is true.

Examples

Solve 2x + 3 = 7 by subtracting 3 from both sides, then dividing both sides by 2, resulting in x = 2.
Solve x - 4 = 9 by adding 4 to both sides, resulting in x = 13.

Special Cases

If the equation is a contradiction (e.g., 2 = 3), there is no solution.
If the equation is an identity (e.g., 2x = 2x), the equation is true for all values of x.

Word Problems

Read the problem and identify the unknown variable.
Write an equation based on the information given.
Solve the equation using the steps above.
Check the solution to ensure it makes sense in the context of the problem.

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Description

Learn how to solve linear equations in one variable by isolating the variable x on one side of the equation. Steps include adding or subtracting constants and multiplying or dividing by coefficients.