Question

Julie subscribes to a sports magazine and a fashion magazine. She received 24 issues this year. The number of fashion issues is 6 less than twice the number of sports issue. Define the variables, and write a system of equations to find the number of issues in each magazine.

Short answer

There are 10 issues in sports magazine and 14 issues in fashion magazine.

Step-by-step solution

Step 1. State the concept of Linear System with two variables.

A linear system of two equations with two variables is any system that can be written in the form.

$\begin{array}{l}ax+by=p\\ cx+dy=q\end{array}$

Where any of the constants can be zero with the exception that each equation must have at least one variable in it.

Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.

Step 2. Write the equations from the conditions given in the question.

Assume $‘x’$to be the number of issues in sports magazine.

Assume $‘y’$to be the number of issues in fashion magazine.

From the 1st condition,

Total numbers of issues received this year is 24. Therefore,

$x+y=24\dots \left(1\right)$

From the 2nd condition,

The number of fashion issues is 6 less than twice the number of sports issue. Therefore,

$y=2x-6\dots \left(2\right)$

Step 3. Solve the above system of equations by substitution method.

Substitution method:Thesubstitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the linear equation gets transformed into one linear equation with only one variable, which can then easily be solved.

Substitute$y=2x-6$in equation (1) and simplify further to find the value of $‘x’$.

$\begin{array}{c}x+y=24\\ x+(2x-6)=24\\ x+2x-6=24\\ 3x-6=24\\ 3x=24+6\\ 3x=30\\ x=\frac{30}{3}\\ x=10\end{array}$

Substitute$x=10$in equation (2) to find the value of $‘y’$.

$\begin{array}{c}y=2x-6\\ y=2\left(10\right)-6\\ y=20-6\\ y=14\end{array}$

Hence, there are 10 issues in sports magazine and 14 issues in fashion magazine.