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