Linear Equations in One Variable Chapter 15

Questions and Answers

What is a linear equation?

An equality involving variables where the highest power of the variable is 1.

Which of the following is a linear expression?

x^2 + 1

2x + 5 (correct)

y + y^2

1 + z + z^2 + z^3

The equation 2x - 3 = 7 has the solution x = 5.

True

How do you determine if a value is a solution to an equation?

The value must make the left-hand side equal to the right-hand side. Signup and view all the answers

In the equation 2x - 3 = 7, the expression on the left is called the ______.

Left Hand Side (LHS) Signup and view all the answers

In the equation 2x - 3 = 7, the expression on the right is called the ______.

Right Hand Side (RHS) Signup and view all the answers

What is meant by 'transposing' in solving equations?

Moving a term from one side of the equation to the other while changing its sign. Signup and view all the answers

To solve the equation 2x - 3 = x + 2, what is the first step?

Subtract x from both sides Signup and view all the answers

Match the following equations with their solutions:

3x = 2x + 18 = x = 18 5t - 3 = 3t - 5 = t = 1 5x + 9 = 5 + 3x = x = 3 4z + 3 = 6 + 2z = z = 1.5 Signup and view all the answers

What should you do when both sides of an equation have expressions with variables?

Perform the same operations on both sides to solve. Signup and view all the answers

What is the solution to the equation 2(x - 1) = 3x - 5?

x = 2 Signup and view all the answers

The highest power of the variable in a linear equation is ______.

1 Signup and view all the answers

Study Notes

Introduction to Linear Equations in One Variable

Algebraic expressions include terms like 5x, 3x + y, and x² + 1, where some contain multiple variables.
Linear equations only involve one variable and have a highest power of 1, e.g., 2x, 3y – 7.
Equations feature an equality sign (=), distinguishing them from expressions.

Understanding Algebraic Equations

An algebraic equation comprises a left-hand side (LHS) and a right-hand side (RHS).
For example, in the equation 2x – 3 = 7, the LHS is 2x – 3, and RHS is 7.
Values that satisfy the equation are termed solutions; for instance, x = 5 is a solution for 2x – 3 = 7.

Solving Linear Equations

The balance of an equation must be maintained, applying the same mathematical operation to both sides.
Equations can have variables on both sides, e.g., 2x – 3 = x + 2, requiring manipulations to isolate the variable.

Examples of Solving Linear Equations

To solve the equation 2x – 3 = x + 2:
- Transpose terms to isolate x, yielding x = 5.
For more complex equations like 5x + 3/2 = x – 14:
- Multiply through by the denominator to eliminate fractions and isolate x accordingly, arriving at x = -5.

Reducing Equations to Simpler Forms

Example: Solve (6x + 1)/3 + 1 = (x - 3)/6:
- Use the least common multiple (LCM) to simplify to a standard form.
The goal is to transform complex equations by opening brackets and combining like terms, leading to simpler equations for easier solving.

Key Points to Remember

Linear equations in one variable are fundamental to algebra, featuring only one variable raised to the first power.
Solutions require testing values for equality on both sides of the equation.
Simplification may involve clearing fractions, combining like terms, and transposing variables across the equation.
Linear equations are powerful tools for solving practical problems involving various scenarios, such as finance, geometry, and age-related problems.

Description

Test your understanding of linear equations in one variable with this quiz from Chapter 15. Dive into the different forms and solutions of linear equations, building on your previous knowledge of algebraic expressions. Perfect for solidifying your grasp of key concepts!