Algebraic Thinking Quiz

1) What is the value of x in the equation 3x + 5 = 20?

5

15

10

7.5

To solve for x in the equation 3x + 5 = 20, begin by isolating x. This is done by subtracting 5 from both sides, which simplifies the equation to 3x = 15. Following this, divide both sides by 3 to fully isolate x, resulting in x = 5. This process not only solves the equation but also illustrates a core principle of algebra: maintaining balance by performing equivalent operations on both sides of the equation, which is critical in correctly solving for any variable.

Explanation

To solve for x in the equation 3x + 5 = 20, begin by isolating x. This is done by subtracting 5 from both sides, which simplifies the equation to 3x = 15. Following this, divide both sides by 3 to fully isolate x, resulting in x = 5. This process not only solves the equation but also illustrates a core principle of algebra: maintaining balance by performing equivalent operations on both sides of the equation, which is critical in correctly solving for any variable.

2) Solve for y: 2y - 6 = 8

4

7

3

5

For the equation 2y - 6 = 8, we solve for y by initially removing the subtraction of 6. This is accomplished by adding 6 to both sides, leading to 2y = 14. The next step is to isolate y by dividing both sides by 2, which gives y = 7. This method showcases how using inverse operations (in this case, addition followed by division) can simplify an equation down to its basic variable, making it straightforward to solve.

Explanation

For the equation 2y - 6 = 8, we solve for y by initially removing the subtraction of 6. This is accomplished by adding 6 to both sides, leading to 2y = 14. The next step is to isolate y by dividing both sides by 2, which gives y = 7. This method showcases how using inverse operations (in this case, addition followed by division) can simplify an equation down to its basic variable, making it straightforward to solve.

3) Which expression is equivalent to 2(x + 5)?

2x + 5

2x + 10

X + 10

4x + 10

The expression 2(x + 5) is simplified by distributing the 2 to each term inside the parentheses, yielding 2x + 10. This demonstrates the distributive property, a pivotal algebraic concept that allows us to simplify expressions by ensuring each term within a set of parentheses is multiplied by an external factor, facilitating further algebraic operations.

Explanation

The expression 2(x + 5) is simplified by distributing the 2 to each term inside the parentheses, yielding 2x + 10. This demonstrates the distributive property, a pivotal algebraic concept that allows us to simplify expressions by ensuring each term within a set of parentheses is multiplied by an external factor, facilitating further algebraic operations.

4) Simplify: 3(a + 4)

3a + 12

3a + 4

3a + 7

7a + 4

In simplifying 3(a + 4), apply the distributive property by multiplying 3 with each term inside the parentheses, resulting in 3a + 12. This property is essential in algebra for breaking down expressions into simpler components, making them easier to handle in complex equations or when combining like terms in polynomial expressions.

Explanation

In simplifying 3(a + 4), apply the distributive property by multiplying 3 with each term inside the parentheses, resulting in 3a + 12. This property is essential in algebra for breaking down expressions into simpler components, making them easier to handle in complex equations or when combining like terms in polynomial expressions.

5) What is the solution for x in x/3 = 9?

27

12

3

6

To isolate x in the equation x/3 = 9, multiply both sides by 3. This action removes the division, simplifying it to x = 27. This example underscores the effectiveness of using multiplication to cancel out division in an equation, a common technique in algebra to simplify and solve equations involving fractions.

Explanation

To isolate x in the equation x/3 = 9, multiply both sides by 3. This action removes the division, simplifying it to x = 27. This example underscores the effectiveness of using multiplication to cancel out division in an equation, a common technique in algebra to simplify and solve equations involving fractions.

6) If y - 3 = 12, what is y?

9

15

10

14

For the equation y - 3 = 12, add 3 to both sides to counteract the subtraction, resulting in y = 15. This illustrates how straightforward addition can balance an equation by eliminating subtraction, providing a clear path to isolating and solving for the variable, a basic yet powerful algebraic technique.

Explanation

For the equation y - 3 = 12, add 3 to both sides to counteract the subtraction, resulting in y = 15. This illustrates how straightforward addition can balance an equation by eliminating subtraction, providing a clear path to isolating and solving for the variable, a basic yet powerful algebraic technique.

7) What is the coefficient in the expression 7y + 3?

7

3

10

0

In the expression 7y + 3, the coefficient of y is 7, which is the numerical factor directly multiplying the variable. The number 3 here serves as a constant and does not affect the coefficient of y. Recognizing coefficients is essential as it helps in understanding how variables are influenced in algebraic expressions and plays a crucial role in further manipulations like distribution or combining like terms.

Explanation

In the expression 7y + 3, the coefficient of y is 7, which is the numerical factor directly multiplying the variable. The number 3 here serves as a constant and does not affect the coefficient of y. Recognizing coefficients is essential as it helps in understanding how variables are influenced in algebraic expressions and plays a crucial role in further manipulations like distribution or combining like terms.

8) Find the value of n: n/2 = 6

3

12

6

18

To solve the equation n/2 = 6, multiply both sides by 2. This operation cancels the division, simplifying the equation to n = 12. This approach exemplifies the use of inverse operations to clear fractions or divisors from equations, making it easier to solve for the variable. It demonstrates a critical aspect of algebra: altering the form of an equation to reveal a more manageable structure.

Explanation

To solve the equation n/2 = 6, multiply both sides by 2. This operation cancels the division, simplifying the equation to n = 12. This approach exemplifies the use of inverse operations to clear fractions or divisors from equations, making it easier to solve for the variable. It demonstrates a critical aspect of algebra: altering the form of an equation to reveal a more manageable structure.

9) Solve for z: 5z - 10 = 0

2

10

0

1

In solving 5z - 10 = 0, adding 10 to both sides yields 5z = 10. Following this, divide both sides by 5 to solve for z, giving z = 2. This process uses two fundamental inverse operations—addition and division—to methodically isolate the variable and solve the equation. It's a prime example of breaking down an equation into simpler steps to find the variable's value.

Explanation

In solving 5z - 10 = 0, adding 10 to both sides yields 5z = 10. Following this, divide both sides by 5 to solve for z, giving z = 2. This process uses two fundamental inverse operations—addition and division—to methodically isolate the variable and solve the equation. It's a prime example of breaking down an equation into simpler steps to find the variable's value.

10) If 4x = 16, what is x?

3

6

4

To find x when 4x = 16, divide both sides of the equation by 4 to isolate x. This simple division reduces the equation to x = 4. This example is a straightforward demonstration of how division can effectively isolate a variable in an equation that involves direct multiplication, highlighting a foundational algebra operation.

Explanation

To find x when 4x = 16, divide both sides of the equation by 4 to isolate x. This simple division reduces the equation to x = 4. This example is a straightforward demonstration of how division can effectively isolate a variable in an equation that involves direct multiplication, highlighting a foundational algebra operation.