Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. Given that \((x, y)\) is a solution to the following system of equations, what is the sum of \(x\) and \(y\) ? \(2 x-y=3\) \(4 y=6 x\)

Short answer

The sum of \(x\) and \(y\) is 15.

Step-by-step solution

- Solve for y in terms of x using the second equation

Start with the second equation: \(4y = 6x\). Solve for \(y\) by dividing both sides by 4. This gives you \(y = \frac{6}{4}x\), which simplifies to \(y = \frac{3}{2}x\).

- Substitute y in the first equation

Now take the expression for \(y\) from Step 1 and substitute it into the first equation: \(2x - y = 3\). Replace \(y\) with \(\frac{3}{2}x\) to get \(2x - \frac{3}{2}x = 3\).

- Simplify the equation

Combine the terms on the left side of the equation. You have \(2x\) which is \(\frac{4}{2}x\), so \(\frac{4}{2}x - \frac{3}{2}x = 3\) simplifies to \(\frac{1}{2}x = 3\).

- Solve for x

To clear the fraction, multiply both sides of the equation \(\frac{1}{2}x = 3\) by 2 to find \(x = 6\).

- Find the corresponding y value

Use the expression \(y = \frac{3}{2}x\) from Step 1. Substitute \(x = 6\) into this equation to find \(y = \frac{3}{2} \times 6 = 9\).

- Calculate the sum of x and y

Now that you have \(x = 6\) and \(y = 9\), find their sum: \(x + y = 6 + 9 = 15\).

Key concepts

System of Equations

A system of equations consists of two or more equations that have the same set of unknowns. In simple terms, you are looking to find values for the variables, often represented by \(x\) and \(y\), that make all the equations true simultaneously. In the given problem, we work with a system of two linear equations:

• \(2x - y = 3\)
• \(4y = 6x\)

The goal is to find a pair \((x, y)\) that satisfies both equations at the same time. This involves turning the equations into simpler forms through algebraic manipulation. Understanding how to solve these systems is a fundamental math skill that is frequently tested on the PSAT and other standardized exams. It involves methods like substitution or elimination to find the solution.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this problem, algebra comes into play as we manipulate the two given equations to solve for the unknowns.
In the provided solution, variables are systematically isolated to express one variable in terms of another, which simplifies the problem. Here's how we used algebra in our steps:

• From the second equation, \(4y = 6x\), we solved for \(y\) giving us \(y = \frac{3}{2}x\).
• This expression for \(y\) is then substituted into the first equation to find the value of \(x\).

By using algebraic techniques like substitution, we simplify and solve complex equations with ease. Solid understanding of algebraic principles is key to tackling PSAT math problems effectively.

Problem-Solving Steps

Effective problem-solving involves breaking down the problem into smaller, more manageable parts and proceeding through an organized series of steps. Let's walk through the solution process step-by-step:- **Solve for one variable:** Start by selecting one equation where you can easily isolate one variable. In this case, it was easier to solve for \(y\) from the equation \(4y = 6x\).
- **Substitute the expression:** Once you have \(y = \frac{3}{2}x\), substitute this back into the other equation (\(2x - y = 3\)) to express it in terms of a single variable.
- **Simplify and solve:** Simplify the equation to find the value of the isolated variable. Here, solving \(\frac{1}{2}x = 3\) gives \(x = 6\).
- **Back-substitute to find other variables:** Use the value found to calculate the corresponding value of the other variable. Substitute \(x = 6\) back to find \(y = 9\).
- **Calculate your answer:** Finally, calculate the sum of the values of \(x\) and \(y\), which is the answer required by the problem.These problem-solving steps are systematic and are helpful across various algebraic problems. By applying them, even complex PSAT math issues can become less daunting.