$\begin{array}{l}
3 x-5 y=21 \\
x-4 y=3
\end{array}$
Answer (1)
Solve the second equation for x : x = 4 y + 3 .
Substitute the expression for x into the first equation and solve for y : y = 7 12 .
Substitute the value of y back into the expression for x and solve for x : x = 7 69 .
The solution to the system of equations is x = 7 69 , y = 7 12 .
Explanation
Understanding the Problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously.
Solving for x We will use the substitution method to solve this system. First, we solve the second equation for x in terms of y : x − 4 y = 3 x = 4 y + 3
Substituting x into the First Equation Now, we substitute this expression for x into the first equation: 3 x − 5 y = 21 3 ( 4 y + 3 ) − 5 y = 21
Solving for y Next, we simplify and solve for y :
12 y + 9 − 5 y = 21 7 y + 9 = 21 7 y = 21 − 9 7 y = 12 y = 7 12
Solving for x Now that we have the value of y , we substitute it back into the expression for x :
x = 4 y + 3 x = 4 ( 7 12 ) + 3 x = 7 48 + 3 x = 7 48 + 7 21 x = 7 69
Final Answer Therefore, the solution to the system of equations is x = 7 69 and y = 7 12 .
Examples
Systems of equations are used in various real-world applications, 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. For instance, if a company wants to know how many units of two different products they need to sell to make a profit, they can set up a system of equations to represent their costs and revenues. Solving this system will give them the quantities of each product they need to sell to break even.
