We are tasked with finding the cost of a notebook and a binder individually, given the following system of equations:

• Maria purchased 15 notebooks and 16 binders for $108.
• Mason purchased 2 notebooks and 9 binders for $35.

Let:

• $x$ represent the cost of one notebook (in dollars).
• $y$ represent the cost of one binder (in dollars).

We can write the system of equations as:

$15x + 16y = 108 \quad \text{(Equation 1)}$ $2x + 9y = 35 \quad \text{(Equation 2)}$

Step 1: Solve Equation 2 for one variable

Let’s solve Equation 2 for $x$.

From Equation 2: $2x + 9y = 35$ Solve for $x$: $2x = 35 - 9y$ $x = \frac{35 - 9y}{2} \quad \text{(Equation 3)}$

Step 2: Substitute Equation 3 into Equation 1

Now substitute the expression for $x$ from Equation 3 into Equation 1:

$15x + 16y = 108$

Substitute $x = \frac{35 - 9y}{2}$:

$15\left( \frac{35 - 9y}{2} \right) + 16y = 108$

Distribute the 15: $\frac{15(35 - 9y)}{2} + 16y = 108$

Simplify the numerator: $\frac{525 - 135y}{2} + 16y = 108$

Multiply through by 2 to eliminate the fraction: $525 - 135y + 32y = 216$

Combine like terms: $525 - 103y = 216$

Step 3: Solve for $y$

Now solve for $y$:

$-103y = 216 - 525$ $-103y = -309$ $y = \frac{-309}{-103} = 3$

Thus, the cost of one binder is $y = 3$ dollars.

Step 4: Substitute $y = 3$ into Equation 3

Now substitute $y = 3$ into Equation 3 to find $x$:

$x = \frac{35 - 9(3)}{2}$ $x = \frac{35 - 27}{2}$ $x = \frac{8}{2} = 4$

Thus, the cost of one notebook is $x = 4$ dollars.

Final Answer:

• The cost of a notebook is $4.
• The cost of a binder is $3.

Would you like further details or clarifications?

Here are 5 related questions:

1. How can we solve systems of equations using the elimination method?
2. What happens if we have more than two variables in the system?
3. How can we use matrices to solve systems of equations?
4. What would happen if Maria had bought fewer notebooks?
5. How do we check the solution of a system of equations?

Tip: Always check your solutions by substituting the values back into the original equations to confirm they satisfy both.