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.
Requested by: Ali
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.
Requested by: Ali
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
Requested by: Ahmer
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.
Requested by: Ali
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.
Requested by: Ali
Question 17:
Solve for x: 3x + 5 = 17
Solution:
3x + 5 = 17
3x = 12
x = 4
Requested by: Ali
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
Requested by: Ali
Question 20:
Solve for z: 2(z + 3) = 16
Solution:
2(z + 3) = 16
2z + 6 = 16
2z = 10
z = 5
Requested by: Ali
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
Requested by: Ali
Question 23:
Solve for x: 2(x + 3) = 16
Solution:
2(x + 3) = 16
2x + 6 = 16
2x = 10
x = 5
Requested by: Ali
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
Requested by: Ali
Question 26:
Solve for x: 5x – 7 = 18
Solution:
5x – 7 = 18
5x = 25
x = 5
Requested by: Ali
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
Requested by: Ali
Question 28:
Simplify the expression: 2x^2 + 5x^2 – 3x^2
Solution:
2x^2 + 5x^2 – 3x^2 = 4x^2
Requested by: Ali
Question 29:
Solve for x: 4(x + 2) = 24
Solution:
4(x + 2) = 24
4x + 8 = 24
4x = 16
x = 4
Requested by: Ali
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
Requested by: Ali
Question 31:
Simplify the expression: 2(x + 4) – 3(x – 2)
Solution:
2(x + 4) – 3(x – 2) = 2x + 8 – 3x + 6 = -x + 14
Requested by: Ali
Question 32:
Solve for y: 5 + 3y = 23
Solution:
5 + 3y = 23
3y = 18
y = 6
Requested by: Ali
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
Requested by: Ali
Question 34:
Simplify the expression: 3x^3 – 2x^3 + x^3
Solution:
3x^3 – 2x^3 + x^3 = 2x^3
Requested by: Ali
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