Linear Equations in One Variable Quiz

Questions and Answers

What is the standard form of a linear equation in one variable?

ax + b = 0 (correct)

ax = b

ax - b = 0

x + b = 0

To isolate the variable in a linear equation, you can only use multiplication.

False

What should you do first when you approach a word problem to set up a linear equation?

Identify the variable and determine what it represents.

For the equation $3x - 7 = 2$, the first step to solve for $x$ is to _____ both sides by 1.

add 7 Signup and view all the answers

Match the linear equation with its corresponding solution:

2x + 5 = 11 = x = 3 3y + 5 = 20 = y = 5 4a - 8 = 0 = a = 2 5b + 10 = 25 = b = 3 Signup and view all the answers

When checking the solution of an equation, what method is typically used?

Substitution Signup and view all the answers

In the equation $2x - 4 = 10$, the solution is found by first adding 4 to both sides.

True Signup and view all the answers

In the problem 'Twice a number decreased by four equals ten', what is the equation formed?

2x - 4 = 10 Signup and view all the answers

To solve the equation $3y + 5 = 20$, you first need to _____ both sides by 3.

subtract 5 Signup and view all the answers

What is the final solution for the equation $x + 5 = 12$?

x = 7 Signup and view all the answers

Study Notes

Linear Equations in One Variable

Solving Techniques

Definition: An equation that can be expressed in the form ( ax + b = 0 ), where ( a ) and ( b ) are constants and ( x ) is the variable.
Isolating the Variable:
1. Addition/Subtraction: Add or subtract terms from both sides to isolate ( x ).
2. Multiplication/Division: Multiply or divide both sides by a non-zero constant to solve for ( x ).
Standard Steps:
1. Simplify both sides of the equation.
2. Move all terms involving ( x ) to one side and constant terms to the other side.
3. Combine like terms.
4. Isolate ( x ) using basic algebraic operations.
Examples:
- ( 2x + 5 = 11 ) → ( 2x = 6 ) → ( x = 3 )
- ( 3x - 7 = 2 ) → ( 3x = 9 ) → ( x = 3 )

Word Problems

Identifying Variables: Determine what the variable represents based on the problem context.
Setting Up Equations: Translate the word problem into a linear equation.
- Identify relationships and values described in the problem.
- Use operations to express these relationships mathematically.
Examples:
- "Twice a number decreased by four equals ten."
  - Let ( x ) be the number: ( 2x - 4 = 10 )
- "Three times a number plus five is equal to twenty."
  - Let ( y ) be the number: ( 3y + 5 = 20 )
Solving the Equation: Follow standard solving techniques outlined above after forming the equation.

Checking Solutions

Substitution: Substitute the found solution back into the original equation to verify correctness.
Steps:
1. Take the solution obtained (e.g., ( x = 3 )).
2. Replace ( x ) in the original equation.
3. Simplify and see if both sides are equal.
Examples:
- For ( x = 3 ) in ( 2x + 5 = 11 ):
  - Check: ( 2(3) + 5 = 6 + 5 = 11 ) (True)
- For ( y = 5 ) in ( 3y + 5 = 20 ):
  - Check: ( 3(5) + 5 = 15 + 5 = 20 ) (True)
Conclusion: If both sides are equal after substitution, the solution is correct. If not, re-evaluate the steps taken to find the solution.

Linear Equations in One Variable

Solving Techniques

Linear equations take the form ( ax + b = 0 ), where ( a ) and ( b ) are constants, and ( x ) is the variable.
Isolate ( x ) using two main strategies:
- Addition/Subtraction: Adjust both sides of the equation by adding or subtracting constants.
- Multiplication/Division: Scale both sides with non-zero constants to derive the value of ( x ).
Follow these standard steps to solve:
- Simplify expressions on both sides of the equation.
- Shift all ( x )-related terms to one side, moving constant terms across the equality.
- Combine like terms for clarity and ease.
- Use algebraic operations to isolate ( x ).
Example of solving:
- From ( 2x + 5 = 11 ) to ( 2x = 6 ), resulting in ( x = 3 ).
- Transforming ( 3x - 7 = 2 ) to find ( x = 3 ) after ( 3x = 9 ).

Word Problems

Identify what the variable symbolizes within the problem's context.
Translate the situation into a mathematical equation by:
- Recognizing relationships and quantities described in the text.
- Utilizing appropriate operations to formulate these relationships quantitatively.
Example translations:
- For “Twice a number decreased by four equals ten,” let ( x ) be the number translates to ( 2x - 4 = 10 ).
- The phrase “Three times a number plus five is equal to twenty” translates to ( 3y + 5 = 20 ) when letting ( y ) be the unknown.

Checking Solutions

To verify solutions, substitute the calculated value back into the original equation.
Steps for verification:
- Utilize the solution obtained (e.g., ( x = 3 )).
- Insert this value into the original equation and simplify to check if both sides of the equation are equal.
Example checks:
- Substituting ( x = 3 ) into ( 2x + 5 = 11 ) confirms correctness as ( 2(3) + 5 = 11 ).
- For ( y = 5 ) in the equation ( 3y + 5 = 20 ), checking gives ( 3(5) + 5 = 20 ), affirming it is true.
Conclusion drawn from checks indicates that if equality holds post-substitution, the solution is valid, otherwise re-evaluation of steps is necessary.

Description

Test your understanding of linear equations in one variable! This quiz covers solving techniques, isolating the variable, and applying methods to word problems. Perfect for students looking to strengthen their algebra skills.