Question

Cristiano bought 24 baseball cards for $\mathbf{\backslash (}\mathbf{50}$. One type cost$\mathbf{\backslash )}\mathbf{1}$per card, and the other cost $\mathbf{\$}\mathbf{3}$per card. Define the variables, and write equations to find the number of each type of card he bought. Solve by using elimination.

Short answer

Cristiano bought 11 cards of one type and 13 cards of other type.

Step-by-step solution

Step 1. State the concept of substitution method and elimination method.

The substitution method is used to find the set of solutions for the given system of equations. It is the method in which one variable is first solved in the form of another variable.

So, the value of one variable in the form second variable is put on the second equation. It will give the solution of the one variable. And then solve it for another variable.

The elimination method is the method in which addition or subtraction method is used to get an equation in the one variable.

If the coefficients are opposite, use the addition and when the coefficients are same, use the subtraction.

Step 2. Add or subtract the equations to eliminate one variable.

Let the number of one type card bought as $\$1$per card be $x$and the number of other type card bought as $\$3$per card be $y$.

As per conditions of problem the system of equations thus formed will be:

$x+y=24$ … (1)

$x+3y=50$ … (2)

Note that both the equations have $x$in same sign, therefore, variable $x$can be eliminated if both equations are subtracted.

$\begin{array}{l}x+y=24\\ \underset{¯}{(-)x+3y=50}\\ -2y=-26\text{Dividebothsidesby-2}\\ y=13\end{array}$

Step 3. Solve for x.

Substitute 13 for y into the equation (1) and solve for $x$.

$\begin{array}{c}x+y=24\\ x+\left(13\right)-13=24-13\\ x=11\end{array}$

Therefore, Cristiano bought 11 cards of one type and 13 cards of other type.