Problem 25 - Algebra 2 and Trigonometry

Question

Solve for the variable and check. Each solution is an integer. \((3 a+7)-(a-1)=14\)

Step-by-Step Solution

Answer

The solution is \(a = 3\) and the solution is verified in the original equation.

Step 1: Distribute the Negative Sign

Simplify the expression by distributing the negative sign in the second term: \((3a + 7) - (a - 1) = 3a + 7 - a + 1\) to obtain \(2a + 8 = 14\).

Step 2: Isolate the Variable

Subtract 8 from both sides of the equation to isolate the term with the variable: \(2a = 14 - 8\) which simplifies to \(2a = 6\).

Step 3: Solve for the Variable

Divide both sides of the equation by 2 to solve for \(a\): \(a = \frac{6}{2} = 3\).

Step 4: Verify the Solution

Substitute \(a = 3\) back into the original equation to verify:Starting with the left side:\((3 \times 3 + 7) - (3 - 1)\)Simplify:\(9 + 7 - 2 = 14\)Since both sides equal 14, the solution is verified.

Key Concepts

Integer Solutions

When solving linear equations, the goal is to find integer solutions, which are whole numbers like 1, -3, or 0. An important part of this is to ensure that every operation leads to clear, whole numbers. In our original equation \[(3a + 7) - (a - 1) = 14,\]the solution aims for an integer because equations with integers typically imply that the variables should resolve to integers as well.

• Integers are numbers without fractions or decimals. This means they can be positive like 1, negative like -2, or even zero.
• To maintain integer solutions, ensure all steps simplify properly. Only use operations that maintain integrity, such as adding, subtracting, or multiplying by integers.
• Keeping the equation balanced is key. Whenever you perform an operation on one side, you must do it on the other side to maintain equality, a fundamental rule for both integer and real number solutions.

By adhering to these principles, solutions will remain simple and precise, ultimately reaching the final integer answer without complication.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators like addition, subtraction, multiplication, and division. In the equation \[(3a + 7) - (a - 1) = 14,\]understanding how to manipulate these expressions is crucial for solving the equation. Here, the goal is to simplify and combine like terms.

• First, distribute any signs. Here, you distribute the negative sign across the terms inside the parentheses to get \[3a + 7 - a + 1.\]
• Combine like terms: Combine terms with 'a' together and constants together: \[(3a - a) = 2a\] and \[(7 + 1) = 8.\]
• Simplify step by step: It's vital to deal with one operation at a time, simplify complex parts gradually.

Mastering these steps ensures that you efficiently reach the simplest form of the expression, allowing for easier manipulation and solution finding.

Verifying Solutions

Verifying your solutions is a vital part of the equation-solving process. It ensures the answer you've calculated satisfies the given equation without any discrepancies. Let's consider how to verify the integer solution for our problem.

• Substitute the solution back into the original equation: For our example, substituting \[a = 3\] back into \[(3a + 7) - (a - 1)\] gives \[(3 \times 3 + 7) - (3 - 1).\]
• Simplify both sides independently to ensure they are equivalent: \[9 + 7 - 2 = 14.\]
• Check for equality: The left side of the equation simplifies to 14, which equals the right side of the original equation, confirming that the solution is correct.

Consistently verifying your solutions helps catch errors and builds confidence in your algebraic skills, confirming the solution's accuracy every time.