Question

If \(3 a+b=3\) and \(b / a=3\) what is \(1 / a+3 / b\) ? A. 1.5 B. 4 C. 3 D. 2.5 E. 5

Short answer

4

Step-by-step solution

- Express b in terms of a

Given that \(\frac{b}{a} = 3\), we can express \(b\) as \(b = 3a\).

- Substitute b in the first equation

Substitute \(b = 3a\) into the first equation \(3a + b = 3\): \[3a + 3a = 3\] \[6a = 3\]

- Solve for a

Solve \(6a = 3\) to find \(a\): \[a = \frac{3}{6} = \frac{1}{2}\]

- Determine the value of b

Use \(a = \frac{1}{2}\) to find \(b\): \[b = 3a = 3 \times \frac{1}{2} = \frac{3}{2}\]

- Find the value of \(\frac{1}{a} + \frac{3}{b}\)

Substitute the values of \(a\) and \(b\) into the expression \(\frac{1}{a} + \frac{3}{b}\): \[\frac{1}{a} + \frac{3}{b} = \frac{1}{\frac{1}{2}} + \frac{3}{\frac{3}{2}} = 2 + 2 = 4\]

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. In this exercise, we have two key algebraic equations: 3a + b = 3 and b/a = 3. Solving these equations involves finding the values of variables a and b. It's important to understand that both sides of an equation must balance. This forms the basis for solving for unknowns by performing various algebraic operations such as addition, subtraction, multiplication, and division.

Substitution Method

The substitution method is a popular way to solve systems of equations. It involves solving one equation for one variable and then substituting this value into another equation. This step transforms a system of equations into a single equation with one variable. Here’s how it was applied:

• We started with the equation b/a = 3 and solved for b, getting b = 3a.
• We then substituted b = 3a into the equation 3a + b = 3, making it easier to solve for a.
• Finally, this substitution simplified our equations and helped us find the values of a and b.

Understanding substitution is crucial for solving more complex systems of equations efficiently.

Fraction Operations

Fraction operations are an important part of algebra. We deal with fractions when we need to simplify or manipulate parts of an equation. Key fraction operations include addition, subtraction, multiplication, and division. Here's a breakdown of the steps necessary:

• First, simplify fractions by finding common denominators when needed.
• Multiply and divide fractions by following the basic rules (e.g., for multiplication, multiply the numerators and denominators).
• In our exercise, we converted the expression 1/a and 3/b into simpler forms.

Practicing fraction operations helps improve your ability to work with different mathematical expressions, making it easier to solve algebraic equations.

Expressions Evaluation

Expression evaluation is about finding the numerical value of an algebraic expression once we have values for its variables. By following these steps, you can evaluate expressions more confidently:

• First, find the values of the variables involved. In our exercise, these were a = 1/2 and b = 3/2.
• Substitute these values into the expression you need to evaluate.
• Perform the necessary arithmetic operations to simplify and find the final value.

In the exercise, evaluating the expression 1/a + 3/b involved substituting the known values of a and b and then simplifying the result to 4. Mastering expression evaluation helps in swiftly solving various algebra problems.