Questions: Fourteen gallons of a salt solution consists of 35% salt. It is the result of mixing a 65% solution with a 30% solution. How many gallons of each of the solutions was used? Let x= the number of gallons of the 65% solution and y= the number of gallons of the 30% solution. The corresponding modeling system is x+y=14, 0.65x+0.3y=0.35(14). Solve the system by using the method of substitution.

Solution Steps

To solve the system of equations using the method of substitution, follow these steps:

1. Solve the first equation $x + y = 14$ for one of the variables, say $y$.
2. Substitute the expression for $y$ into the second equation $0.65x + 0.3y = 0.35 \times 14$.
3. Solve the resulting equation for $x$.
4. Substitute the value of $x$ back into the expression for $y$ to find $y$.

We start with the system of equations based on the problem statement: \[ \begin{align*}

1. & \quad x + y = 14 \quad \text{(1)} \\
2. & \quad 0.65x + 0.3y = 0.35 \times 14 \quad \text{(2)} \end{align*} \] Calculating $0.35 \times 14$ gives us $4.9$, so equation (2) can be rewritten as: $$0.65x + 0.3y = 4.9$$

From equation (1), we can express $y$ in terms of $x$: $$y = 14 - x \quad \text{(3)}$$

Substituting equation (3) into equation (2): $$0.65x + 0.3(14 - x) = 4.9$$ Expanding this gives: $$0.65x + 4.2 - 0.3x = 4.9$$ Combining like terms results in: $$0.35x + 4.2 = 4.9$$ Subtracting $4.2$ from both sides: $$0.35x = 0.7$$ Dividing by $0.35$: $$x = 2$$

Now substituting $x = 2$ back into equation (3) to find $y$: $$y = 14 - 2 = 12$$

Final Answer