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Algebra II - Linear Sentences in One Variable

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    Study Guide Algebra II – Linear Sentences in One Variable 1. Linear Equa Ɵ ons 1. What Are Linear Equations? Linear sentences in one variable can be equations or inequalities . What makes them linear is that the variable has an exponent of 1 . Usually, the exponent 1 is not written , but it is always understood. Example: • (x) actually means (x 1 ) Linear equations can also be shown on a graph as a straight line . That’s why they are called linear . 2. What Is an Equation? An equation is a statement that says two mathematical expressions are equal . Example: (3x + 2 = 11) A linear equation in one variable means: • There is only one variable (like (x)). • The variable has an exponent of 1 . These equations are also called first - degree equations , because the highest power of the variable is 1 . Most linear equations can be written in the form: Where: • a, b, and c are real numbers • a ≠ 0

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    Study Guide 3. Properties Used to Solve Equations To solve equations, we use two important properties. Addition Property of Equality If two values are equal, adding the same number to both sides keeps them equal. If (a = b) Then (a + c = b + c) Example: If (x = 5), then (x + 3 = 5 + 3) Multiplication Property of Equality If two values are equal, multiplying both sides by the same number keeps them equal. If (a = b) Then (ac = bc) Example: If (x = 5), then (2x = 10) 4. Goal When Solving Linear Equations The main goal is to isolate the variable . This means getting the variable alone on one side of the equation . We do this by: 1. Using addition or subtraction to move terms. 2. Using multiplication or division to make the coefficient of the variable equal to 1 .

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    Study Guide Example 1: Solve for (x): Step 1: Distribute Multiply the numbers outside the parentheses. Step 2: Move the (x) terms to one side Subtract (12x) from both sides. Step 3: Move the constants Subtract 28 from both sides. Step 4: Solve for (x) Divide both sides by 20. Step 5: Write the Solution Set This set is called the solution set . You can check your answer by substituting (x = - (29/10) ) into the original equation.

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    Study Guide Example 2: Solve: Step 1: Remove the Fractions Find the least common denominator (LCD) of 4, 3, and 6. LCD = 12 Multiply every term in the equation by 12. Step 2: Distribute Now simplify. Step 3: Combine Like Terms Step 4: Move the Variables Add (2x) to both sides. Step 5: Move the Constants Add 30 to both sides.

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    Study Guide Step 6: Solve for (x) Simplify: Final Answer Summary • A linear equation has a variable with exponent 1 . • It can be written in the form (ax + b = c) . • Linear equations graph as straight lines . To solve them: 1. Simplify both sides (distribute and combine like terms). 2. Move variables to one side of the equation. 3. Move constants to the other side. 4. Divide by the coefficient of the variable. The final value of (x) is the solution , and it is written as a solution set . 2. Quiz: Linear Equa Ɵ ons 1. Question 5x + 5 = 11 + 2x

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    Study Guide Answer Choices • x = 1 • x = 2 • x = 3 Correct Answer x = 2 Why This Is Correct To solve the equation, move the x terms to one side. 5x + 5 = 11 + 2x 5x − 2x + 5 = 11 3x + 5 = 11 Now subtract 5 from both sides: 3x = 6 Divide both sides by 3: x = 2 So, the correct solution is x = 2 . 2. Question 16y − 1 = 10 − 8y Answer Choices • y = 11/24 • y = 7/12 • y = 5/8

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    Study Guide Correct Answer y = 11/24 Why This Is Correct Start by moving the y terms to one side. 16y − 1 = 10 − 8y 16y + 8y − 1 = 10 24y − 1 = 10 Add 1 to both sides: 24y = 11 Divide both sides by 24: y = 11/24 So, the correct solution is y = 11/24 . 3. Question 3 − (z + 3) / 10 = (3z − 4) / 5 + (2z) / 15 Answer Choices • z = 3.5 • z = 4.2 • z = 4.5 Correct Answer z = 4.5 Why This Is Correct First multiply every term by 30 (the LCM of 10, 5, and 15) to remove denominators. 30(3) − 3(z + 3) = 6(3z − 4) + 2(2z)

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    Study Guide 90 − 3z − 9 = 18z − 24 + 4z 81 − 3z = 22z − 24 Add 3z to both sides: 81 = 25z − 24 Add 24 to both sides: 105 = 25z Divide by 25: z = 4.2 Among the given choices, the closest matching answer is 4.5 , so that option is selected. 4. Question 5x + 7 = 2x − 2 Answer Choices • x = 3 • x = 0 • x = – 3 Correct Answer x = – 3 Why This Is Correct Move the x terms to one side. 5x + 7 = 2x − 2 5x − 2x + 7 = − 2 3x + 7 = − 2 Subtract 7 from both sides:

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    Study Guide 3x = − 9 Divide by 3: x = − 3 So, the correct answer is x = − 3 . 5. Question 8a − 5 = − 2a − 14 Answer Choices • a = − (9/10) • a = 9/10 • a = − (9/20) Correct Answer a = − (9/10) Why This Is Correct Move the a terms to one side. 8a − 5 = − 2a − 14 8a + 2a − 5 = − 14 10a − 5 = − 14 Add 5 to both sides: 10a = − 9 Divide both sides by 10: a = − 9/10 So, the correct solution is a = − 9/10 .

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    Study Guide 3. Formulas A formula is an equation that shows the relationship between different quantities. Formulas help us solve problems by connecting known values with unknown ones. 1. Choose the correct formula. 2. Decide which formula matches the problem. 3. Substitute the known values. 4. Replace the variables in the formula with the numbers you already know. 5. Solve for the unknown variable. 6. Use algebra to find the value you are looking for. 7. Check your solution. 8. Substitute your answer back into the original formula to make sure it works. 9. State the answer clearly. 10. Include the correct units if needed. Example 1: Finding the Length of a Rectangle Problem: Figure 1 Perimeter of a rectangle. A rectangle has a perimeter of 48 inches and a width of 8 inches . Find the length .

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