**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) Ã· 2 – 1**

**Step 1: Simplify the expression inside the parentheses: 3(6) Ã· 2 – 1.**

**Step 2: Perform the division: 18 Ã· 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.**

**Requested by: Ali**

**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**

**Requested by: Ali**

**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.**

**Requested by: Ali**

**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.**

**Requested by: Ali**

**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**

**Requested by: Ali**

**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**

**Requested by: Ali**

**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**

**Requested by: Ali**

**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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