Question

If \((a, b)\) is a solution to the system of equations below, what is the value of \(a\) ? (A) \(2 a-\frac{1}{2} b=4\) (B) \(3 a+b=6\)

Short answer

The value of \(a\) is 2.

Step-by-step solution

Identify the System of Equations

We have two equations: \(2a - \frac{1}{2}b = 4\) and \(3a + b = 6\). Our goal is to find the value of \(a\) by solving this system of equations.

Solve One Equation for One Variable

From equation (B) \(3a + b = 6\), let's solve for \(b\): \(b = 6 - 3a\).

Substitute into the Other Equation

Substitute \(b = 6 - 3a\) from Step 2 into equation (A): \(2a - \frac{1}{2}(6 - 3a) = 4\).

Simplify and Solve for a

Expand and simplify the equation: \(2a - \frac{6}{2} + \frac{3}{2}a = 4\). Combine like terms to get \(2a + \frac{3}{2}a = 4 + 3\). This simplifies to \(\frac{4a}{2} + \frac{3a}{2} = 7\). Therefore, \(\frac{7a}{2} = 7\). Multiply by 2 to clear the fraction: \(7a = 14\), then divide by 7 to find \(a\): \(a = 2\).

Validate Your Solution

Substitute \(a = 2\) back into equation (B) to check our solution: \(3(2) + b = 6\), which gives \(6 + b = 6\). Thus, \(b = 0\). Substitute both values into equation (A) to confirm \(2(2) - \frac{1}{2}(0) = 4\), which holds true, confirming \(a = 2\) is correct.

Key concepts

Algebra

Algebra is like a magical toolkit in mathematics. It helps us to express real-world problems with letters and symbols. These symbols often represent unknowns that we can figure out using mathematical operations.
In the given problem, we have a system of equations. Each equation is an algebraic way to describe a relationship. When we say "system of equations," we mean two or more equations that share the same variables.
Here, the equations involve the variables \( a \) and \( b \). When solving algebraic problems, our main goal is to find the value of these variables. This is done by manipulating the equations, using operations like addition, subtraction, multiplication, and division, to isolate one variable and solve for its value.
When two equations are given together like this, they provide unique clues that lead us to the exact solution for the unknowns.

Solving Equations

Solving equations is all about finding values that make the equation true. In our system, we need to solve for \( a \) and \( b \). This involves several steps.
First, we solve one of the equations for one of the variables, like solving for \( b \) in terms of \( a \) as we did in the solution. This is often called "substitution."

• We found \( b = 6 - 3a \) from equation (B).
• Next, we substituted this expression for \( b \) back into equation (A).

By doing so, we turned a system of two equations into a single equation with only one variable \( a \). This simplifies matters significantly.
Finally, we solve this new equation to find the numerical value of \( a \). The process involves expanding, simplifying the expressions, and performing operations to isolate \( a \).
This step-by-step approach ensures that we carefully handle each part of the system, ultimately leading to the correct solution for the variable of interest.

PSAT Math

The PSAT math section often includes problems like systems of equations, making it essential for students to be comfortable with algebraic manipulation.
This problem type tests your ability to:

• Understand and interpret the given equations.
• Execute an effective strategy to reach the solution.
• Apply algebraic techniques such as substitution and elimination to find the unknowns.

On the PSAT, it's crucial to approach these problems one step at a time, as we did here by first isolating a variable. Recognizing which technique to use is a skill that improves with practice.
These exercises not only test mathematical ability but also logical thinking and problem-solving skills, which are vital for success in both exams and real-world applications. Practicing with systems of equations not only helps on standardized tests like the PSAT but also in developing strong foundational math skills.