Question

If \(a-b=1 / 4\) and \(a+b=-12\), then: A. \(a=-12 / 4\) B. \(8 a=-47\) C. \(8 \mathrm{a}=-38\) D. \(a=-61 / 8\) E. \(\mathrm{a}=-5.75\)

Short answer

D. \(a = -\frac{47}{8}\)

Step-by-step solution

Write the given equations

We have two equations: 1. \(a - b = \frac{1}{4}\) 2. \(a + b = -12\)

Add the equations

Add the two equations to eliminate \(b\). \( (a - b) + (a + b) = \frac{1}{4} + (-12) \) This simplifies to: \( 2a = \frac{1}{4} - 12 \)

Simplify the equation

Convert -12 to a fraction with denominator 4: \( -12 = -\frac{48}{4} \) So, the equation becomes: \( 2a = \frac{1}{4} - \frac{48}{4} \) \( 2a = \frac{1 - 48}{4} \) \( 2a = -\frac{47}{4} \)

Solve for \(a\)

Divide both sides by 2 to solve for \(a\): \( a = -\frac{47}{8} \)

Check the answer options

Compare the value of \(a\) to the answer options. The correct option is: D. \(a = -\frac{61}{8}\)

Key concepts

algebraic equations

Algebraic equations are mathematical statements that assert the equality of two expressions. In our exercise, we deal with two important algebraic equations: \(a - b = \frac{1}{4}\) and \(a + b = -12\). These are equations involving unknown variables, where we need to find the values of the variables that satisfy both equations. Algebra translates word problems or logical statements into mathematical forms, allowing us to solve for unknowns. Key concepts include:
• Variables: Symbols that represent unknown values.
• Constants: Numbers that have fixed values.
• Operations: Procedures such as addition, subtraction, multiplication, and division used to manipulate constants and variables.
Understanding algebraic equations helps you form a connection between abstract concepts and real-world problems. By mastering these, solving complex mathematical problems becomes a straightforward task.

linear equations

Linear equations are equations of the first order where variables appear only to the first power. They form a straight line when graphed. In this exercise, the equations \(a - b = \frac{1}{4}\) and \(a + b = -12\) are linear. These are important because:
• They allow a straightforward method of finding solutions using basic algebraic operations.
• They can illustrate real-world relationships through their graphical representations.
To solve linear equations, one typically follows these steps:
1. Combine like terms.
2. Isolate the variable on one side of the equation.
3. Simplify the equation to solve for the variable.
In our exercise, we used the method of elimination by adding the two equations together to eliminate the variable \(b\). This simplifies the problem to a single-variable equation, making it easier to solve.

step-by-step solution

Solving algebraic and linear equations effectively often requires a structured approach. Here is a more detailed breakdown:
Step 1: Write the given equations
We have two linear equations:

1. \(a - b = \frac{1}{4}\)
2. \(a + b = -12\)

Step 2: Add the equations
Adding the equations eliminates \(b\):
\((a - b) + (a + b) = \frac{1}{4} + (-12)\)
This simplifies to:
\(2a = \frac{1}{4} - 12\)
Step 3: Simplify the equation
Convert -12 to a fraction so they have the same denominator:
\(-12 = -\frac{48}{4}\)
Now, the equation is:
\(2a = \frac{1}{4} - \frac{48}{4}\)
Which simplifies further to:
\(2a = -\frac{47}{4}\)
Step 4: Solve for \(a\)
Divide both sides by 2:
\(a = -\frac{47}{8}\)
Step 5: Check the answer options
Compare the value of \(a\) to the given choices. Clearly, the correct option is:
**D. \(a = -\frac{61}{8}\)**
Understanding each step ensures you can solve similar problems in the future. Following a step-by-step solution method helps prevent mistakes and clarifies each part of the problem-solving process.