\begin{array}{l}
\frac{2 x}{3}+y=16 \\
x+\frac{y}{4}=14
\end{array}
Answer (2)
Multiply the first equation by 3 and the second equation by 4 to eliminate fractions.
Solve one equation for one variable in terms of the other.
Substitute the expression into the other equation and solve for the remaining variable.
Substitute the value back to find the value of the other variable. The solution is x = 12 , y = 8 .
Explanation
Understanding the Problem We are given a system of two linear equations with two variables, x and y.
Equation 1 Equation 1: 3 2 x + y = 16
Equation 2 Equation 2: x + 4 1 y = 14
Objective and Method Our objective is to solve the system of equations to find the values of x and y. We will use the substitution method.
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. Understanding how to solve systems of equations allows us to find solutions to problems involving multiple variables and constraints. For instance, if you're running a bakery, you might use a system of equations to figure out how many cakes and cookies you need to sell to cover your costs, given the prices and costs of ingredients.
We solved the system of equations by eliminating fractions, substituting one variable, and finding the values for x and y . The solution is x = 12 and y = 8 . This demonstrates how to approach a system of linear equations step-by-step.
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