Question

Solve \(\left\\{\begin{array}{r}3 x+y=5 \\ 2 x-3 y=7\end{array}\right.\).

Short answer

x = 2, y = -1

Step-by-step solution

- Write Down the System of Equations

The system of linear equations is given as: 1) 3x + y = 5 2) 2x - 3y = 7

- Multiply the First Equation by 3

To eliminate y, multiply the entire first equation by 3 to make the coefficients of y in both equations equal: 1) 3(3x + y) = 3(5) This simplifies to: 9x + 3y = 15

- Add the Equations

Add the modified first equation (9x + 3y = 15) and the original second equation (2x - 3y = 7): (9x + 3y) + (2x - 3y) = 15 + 7 This simplifies to: 11x = 22 Solve for x: \[ x = \frac{22}{11} = 2 \]

- Substitute x Back into One of the Original Equations

Substitute x = 2 back into the first original equation (3x + y = 5): 3(2) + y = 5 This simplifies to: 6 + y = 5 Solve for y: \[ y = 5 - 6 = -1 \]

- State the Solution

The solution to the system of equations is x = 2 and y = -1

Key concepts

system of equations

A system of equations consists of two or more equations with multiple variables that are solved together because they share common variables. In this exercise, our system is: 3x + y = 5 and 2x - 3y = 7.

Systems of equations can be solved using various methods. The two most common methods are the substitution method and the elimination method. Both methods aim to simplify the system to find the values of the variables that satisfy all the equations in the system.

substitution method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This approach is useful when one equation is easily solvable for one of the variables.

Here’s how it works:

• Solve one of the equations for one variable in terms of the others.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.
• Substitute the found value back into the original equation to find the other variable.

In this exercise, we could use the substitution method by solving the first equation for y: y = 5 - 3x. Then, substitute this expression into the second equation 2x - 3(5 - 3x) = 7. This involves more steps, so the elimination method was chosen instead.

elimination method

The elimination method, also known as the addition method, involves adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of one of the variables are opposites or can be made opposites.

Here’s a step-by-step of the elimination method used in this exercise:

• Write down the system of equations.
• Multiply one or both equations so that the coefficients of one of the variables are the same (or opposites).
• Add or subtract the equations to eliminate one variable.
• Solve the resulting single-variable equation.
• Substitute back into one of the original equations to find the other variable.

In our example, we multiplied the first equation by 3 to match the coefficients of y. Then, adding the equations eliminated y and allowed us to solve for x first, and subsequently for y.