Question

If \(x / 2+y / 3=2,2 x+3 y=13\), then \(y\) is: (A) \(13 / 3\) (B) \(8 / 3\) (C) 5 (D) \(3 / 5\) (E) 3

Short answer

The value of y is 3 (Option E).

Step-by-step solution

- Write Down the Equations

We have two equations:1) \(\frac{x}{2} + \frac{y}{3} = 2\)2) \(2x + 3y = 13\)

- Eliminate Decimals and Fractions

Multiply the first equation by 6 to eliminate the fractions. This simplifies equation (1).\[6 \times \frac{x}{2} + 6 \times \frac{y}{3} = 6 \times 2\]\[3x + 2y = 12\]

- Solve for One Variable

We now have the simplified system of equations:1) \(3x + 2y = 12\)2) \(2x + 3y = 13\)Multiply equation (1) by 3 and equation (2) by 2 to align the coefficients of y:\[3(3x + 2y) = 3(12)\]\[2(2x + 3y) = 2(13)\]This results in:\[9x + 6y = 36\]\[4x + 6y = 26\]

- Subtract the Equations

Subtract the second equation from the first to eliminate y:\[(9x + 6y) - (4x + 6y) = 36 - 26\]\[5x = 10\]\[x = 2\]

- Substitute Back to Find y

Use the value of x in one of the original equations to find y. Substitute x into equation (1):\[3(2) + 2y = 12\]\[6 + 2y = 12\]\[2y = 6\]\[y = 3\]

- Verify the Solution

Substitute both x and y back into both original equations to verify the solution.For equation (1):\[\frac{2}{2} + \frac{3}{3} = 2\]\[1 + 1 = 2\]For equation (2):\[2(2) + 3(3) = 13\]\[4 + 9 = 13\]

Key concepts

systems of equations

A system of equations is a collection of two or more equations with the same set of variables. In our exercise, we have the equations:

• \(\frac{x}{2} + \frac{y}{3} = 2\)
• \(\text{2x + 3y = 13}\)

The goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. By solving these equations together, we determine the point where the two equations intersect when graphed on a coordinate plane. This point represents the unique solution for both variables.

solving linear equations

Solving linear equations involves finding the values of variables that make the equation true. A linear equation is one where each term is either a constant or the product of a constant and a single variable.

The equations in our problem are linear:

• \(\frac{x}{2} + \frac{y}{3} = 2\)
• \(2x + 3y = 13\)

To solve a linear equation, you can isolate the variable on one side of the equation through algebraic operations such as addition, subtraction, multiplication, or division. Each step should simplify the equation until the solution is clear.

substitution method

The substitution method is one way to solve a system of equations. It involves solving one equation for one variable and then substituting that expression into the other equation.

Here’s how it works:

• Solve one of the equations for one variable in terms of the other variable.
• Substitute this expression into the other equation.
• Solve the resulting equation to find the value of one variable.
• Plug this value back into the equation used in step one to find the other variable.

In our example, this could involve solving the simplified equation \(3x + 2y = 12\) for \(x\) or \(y\) first. However, we used the elimination method, as outlined in the steps.

elimination method

The elimination method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve the remaining equation.

Let’s break it down:

• First, align the equations so both are simplified, like we did by eliminating fractions in the first equation.
• Next, multiply one or both equations so that when added or subtracted, one variable cancels out. In our problem, we multiplied the equations to align the coefficients of \(y\).
• Then, subtract the equations to eliminate the \(y\) variable and solve for \(x\).
• Finally, substitute the \(x\) value back into one of the original equations to find \(y\).

In our problem, we simplified to get \(3x + 2y = 12\) and then used elimination to find \(x = 2\), and subsequently \(y = 3\). This is an efficient method when the equations are set up nicely for elimination.