Basic algebraic concepts (solving for an unknown, writing and evaluating algebraic expressions)

Solving an equation for an unknown:

Step 1: Identify the unknown variable in the equation.

Step 2: Use algebraic operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the equation.

Step 3: Simplify the equation by performing the necessary operations.

Step 4: Check your answer by substituting it back into the original equation and verifying that both sides of the equation are equal.

###### For Example:

consider the equation: 3x – 5 = 10

Step 1: The unknown variable in this equation is x.

Step 2: Add 5 to both sides of the equation to isolate the variable: 3x = 15.

Step 3: Divide both sides of the equation by 3 to solve for x:

x = 5.

Step 4: Check the answer by substituting x = 5 back into the original equation:

3(5) – 5 = 10, which is true. So the solution to the equation is x = 5.

###### Writing and evaluating an algebraic expression:

###### Step 1: Identify the variables and constants in the expression.

Step 2: Determine the operations involved (addition, subtraction, multiplication, division).

Step 3: Simplify the expression by performing the necessary operations.

Step 4: Evaluate the expression by substituting numerical values for the variables.

###### For Example:

consider the expression: 2x + 3 when x = 4.

Step 1: The variable is x and the constant is 3.

Step 2: The operation involved is addition.

Step 3: Simplify the expression by substituting x = 4:

2(4) + 3 = 11.

Step 4: So the value of the expression is 11 when x = 4.

###### Writing and simplifying algebraic expressions:

###### Algebraic expressions can also be simplified by combining like terms.

Like terms are terms that have the same variables raised to the same powers.

###### For example, the terms 3x and -2x are like terms because they both have x as a variable raised to the power of 1.

To simplify expressions, we can add or subtract the coefficients (the numbers in front of the variables) of the like terms.

###### For Example:

Simplify the expression: 5x + 2x – 3y + y

###### Step 1: Combine the like terms: 5x + 2x = 7x,

and -3y + y = -2y.

Step 2: Substitute the combined like terms back into the expression: 7x – 2y.

So the simplified expression is 7x – 2y.

###### Order of Operations

###### When simplifying algebraic expressions, it’s important to follow the order of operations, which is a set of rules that tells us which operations to perform first.

The order of operations is PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

###### For Example:

Evaluate the expression: 3(4 + 2) ÷ 2 – 1

Step 1: Simplify the expression inside the parentheses: 3(6) ÷ 2 – 1.

Step 2: Perform the division: 18 ÷ 2 = 9.

Step 3: Perform the subtraction: 9 – 1 = 8.

So the value of the expression is 8.

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Step by Step Solutions to PEDMAS Problems

###### Question 1:

Simplify the expression: 2x + 3y – 2x – y

Solution:

To simplify the expression, we combine the like terms:

2x – 2x = 0x, which simplifies to 0, so we can eliminate the x terms

3y – y = 2y

Therefore, the simplified expression is: x + 2y.

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###### Question 2:

5(x – 3) = 20 Solve for x:

Solution:

To solve for x, we first use the distributive property to expand the left side of the equation:

5(x – 3) = 20

5x – 15 = 20

Next, we add 15 to both sides of the equation to isolate the variable term:

5x = 35

Finally, we divide both sides of the equation by 5 to solve for x:

x = 7

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###### Question 3:

Evaluate the expression: (2a + b)^2 when a = 3 and b = 4

Solution:

To evaluate the expression, we substitute the given values of a and b into the expression, and then simplify:

(2a + b)^2 = (2(3) + 4)^2 = (6 + 4)^2 = 10^2 = 100

Therefore, the value of the expression is 100.

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###### Question 4:

Simplify the expression: 2(x + 3) – 3(2x – 1)

Solution:

To simplify the expression, we first use the distributive property to expand the second term:

2(x + 3) – 3(2x – 1) = 2x + 6 – 6x + 3

Next, we combine the like terms:

-4x + 9

Therefore, the simplified expression is -4x + 9.

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###### Question 5:

3x – 2 = x + 8. Solve for x:

Solution:

To solve for x, we first isolate the variable term by adding 2 and subtracting x from both sides of the equation:

3x – 2 – x = 8 + x – x

2x – 2 = 8

Next, we add 2 to both sides of the equation to isolate the variable term:

2x = 10

Finally, we divide both sides of the equation by 2 to solve for x:

x = 5

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###### Question 6:

Evaluate the expression: 4x^2 + 6x – 2 when x = -2

Solution:

To evaluate the expression, we substitute the given value of x into the expression, and then simplify:

4(-2)^2 + 6(-2) – 2 = 16 – 12 – 2 = 2

Therefore, the value of the expression is 2.

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###### Question 7:

Simplify the expression: 3(2x – 4) – 2(x – 3)

Solution:

To simplify the expression, we first use the distributive property to expand both terms:

3(2x – 4) – 2(x – 3) = 6x – 12 – 2x + 6

Next, we combine the like terms:

4x – 6

Therefore, the simplified expression is 4x – 6.

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###### Question 8:

2(3x – 1) = 10 – 2x. Solve for x

Solution:

To solve for x, we first use the distributive property to expand the left side of the equation:

2(3x – 1) = 10 – 2x

6x – 2 = 10 – 2x

Next, we add 2x and 2 to both sides of the equation to isolate the variable term:

8x = 12

Finally, we divide both sides of the equation by 8 to solve for x:

x = 3/4 or 0.75

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###### Question 9:

Evaluate the expression: 2x^2 – 5x + 3 when x = 2

Solution:

To evaluate the expression, we substitute the given value of x into the expression, and then simplify:

2(2)^2 – 5(2) + 3 = 8 – 10 + 3 = 1

Therefore, the value of the expression is 1.

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###### Question 10:

Simplify the expression: (4x^2 + 3x – 2) – (x^2 – 5x + 4)

Solution:

To simplify the expression, we use the distributive property to distribute the negative sign to the second term:

(4x^2 + 3x – 2) – (x^2 – 5x + 4) = 4x^2 + 3x – 2 – x^2 + 5x – 4

Next, we combine the like terms:

3x^2 + 8x – 6

Therefore, the simplified expression is 3x^2 + 8x – 6.

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###### Question 11:

5(x – 4) = 10x + 15. Solve for x

Solution:

To solve for x, we first use the distributive property to expand the left side of the equation:

5x – 20 = 10x + 15

Next, we subtract 5x from both sides of the equation to isolate the variable term:

-20 = 5x + 15

Then, we subtract 15 from both sides of the equation:

-35 = 5x

Finally, we divide both sides of the equation by 5 to solve for x:

x = -7

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###### Question 12:

Evaluate the expression: 3x^2 – 2x + 1 when x = -1

Solution:

To evaluate the expression, we substitute the given value of x into the expression, and then simplify:

3(-1)^2 – 2(-1) + 1 = 3 + 2 + 1 = 6

Therefore, the value of the expression is 6.

Requested by: Ali

###### Question 13:

Simplify the expression: 2x(3x + 4) + 5x(2x – 3)

Solution:

To simplify the expression, we use the distributive property to expand both terms:

2x(3x + 4) + 5x(2x – 3) = 6x^2 + 8x + 10x^2 – 15x

Next, we combine the like terms:

16x^2 – 7x

Therefore, the simplified expression is 16x^2 – 7x.

Requested by: Ahmer

###### Question 14:

Solve for x: (x – 3)/4 + 2 = (2x + 1)/3

Solution:

To solve for x, we first multiply both sides of the equation by the least common multiple of the denominators, which is 12:

3(x – 3) + 24 = 4(2x + 1)

Next, we distribute the 3 and 4 to their respective terms:

3x – 9 + 24 = 8x + 4

Then, we combine the like terms:

3x + 15 = 8x + 4

Next, we subtract 3x from both sides of the equation to isolate the variable term:

15 = 5x + 4

Finally, we subtract 4 from both sides of the equation to solve for x:

x = 2.2

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###### Question 15:

Evaluate the expression: (x^2 + 3x + 2)/(x + 2) when x = -3

Solution:

To evaluate the expression, we substitute the given value of x into the expression, and then simplify:

(-3)^2 + 3(-3) + 2/(-3 + 2) = 4/-1 = -4

Therefore, the value of the expression is -4.

Requested by: Ali

###### Question 16:

Simplify the expression: (2x – 1)(x + 3) + 4(x – 2)(x + 1)

Solution:

To simplify the expression, we first use the distributive property to expand both terms:

(2x – 1)(x + 3) + 4(x – 2)(x + 1) = 2x^2 + 5x – 3 + 4x^2 – 4x – 8

Next, we combine the like terms:

6x^2 + x – 11

Therefore, the simplified expression is 6x^2 + x – 11.

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###### Question 17:

Solve for x: 3x + 5 = 17

Solution:

3x + 5 = 17

3x = 12

x = 4

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###### Question 18:

Solve for y: 6y – 8 = 22

Solution:

6y – 8 = 22

6y = 30

y = 5

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###### Question 19:

Simplify the expression: 4(x + 3) – 2x

Solution:

4(x + 3) – 2x = 4x + 12 – 2x = 2x + 12

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###### Question 20:

Solve for z: 2(z + 3) = 16

Solution:

2(z + 3) = 16

2z + 6 = 16

2z = 10

z = 5

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###### Question 21:

Evaluate the expression: 3x – 5y when x = 4 and y = 2

Solution:

3x – 5y = 3(4) – 5(2) = 2

Requested by: Ali

###### Question 22:

Simplify the expression: 2x + 4y – 3x – 5y

Solution:

2x – 3x + 4y – 5y = -x – y

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###### Question 23:

Solve for x: 2(x + 3) = 16

Solution:

2(x + 3) = 16

2x + 6 = 16

2x = 10

x = 5

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###### Question 24:

Evaluate the expression: 4x^2 – 3y^2 when x = 2 and y = 3

Solution:

4x^2 – 3y^2 = 4(2)^2 – 3(3)^2 = 4(4) – 3(9) = -3

Requested by: Ali

###### Question 25:

Simplify the expression: 3x^2 + 2x^2 – x^2

Solution:

3x^2 + 2x^2 – x^2 = 4x^2

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###### Question 26:

Solve for x: 5x – 7 = 18

Solution:

5x – 7 = 18

5x = 25

x = 5

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###### Question 27:

Evaluate the expression: 2x^3 – 4y^2 when x = -1 and y = 2

Solution:

2x^3 – 4y^2 = 2(-1)^3 – 4(2)^2 = -28

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###### Question 28:

Simplify the expression: 2x^2 + 5x^2 – 3x^2

Solution:

2x^2 + 5x^2 – 3x^2 = 4x^2

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###### Question 29:

Solve for x: 4(x + 2) = 24

Solution:

4(x + 2) = 24

4x + 8 = 24

4x = 16

x = 4

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###### Question 30:

Evaluate the expression: 3x^2 – 2xy + y^2 when x = 2 and y = 3

Solution:

3x^2 – 2xy + y^2 = (3)(2)^2 – 2(2)(3) + (3)^2 = 9

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###### Question 31:

Simplify the expression: 2(x + 4) – 3(x – 2)

Solution:

2(x + 4) – 3(x – 2) = 2x + 8 – 3x + 6 = -x + 14

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###### Question 32:

Solve for y: 5 + 3y = 23

Solution:

5 + 3y = 23

3y = 18

y = 6

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###### Question 33:

Evaluate the expression: 2x^2 – 3xy + y^2 when x = 3 and y = 4

Solution:

2x^2 – 3xy + y^2 = 2(3)^2 – 3(3)(4) + (4)^2 = -2

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###### Question 34:

Simplify the expression: 3x^3 – 2x^3 + x^3

Solution:

3x^3 – 2x^3 + x^3 = 2x^3

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