\begin{array}{l}2 x-3 y=4 \\ 4 x-y=13\end{array}
Answer (1)
Multiply the first equation by 2: 4 x − 6 y = 8 .
Subtract the modified first equation from the second equation: 5 y = 5 .
Solve for y : y = 1 .
Substitute y = 1 into the first equation and solve for x : x = 2 7 .
The solution is x = 2 7 , y = 1 .
Explanation
Analyze the problem and choose a method We are given a system of two linear equations:
2 x − 3 y = 4 4 x − y = 13
Our goal is to find the values of x and y that satisfy both equations. We can use either substitution or elimination method.
Multiply the first equation by 2 Let's use the elimination method. Multiply the first equation by 2:
2 ( 2 x − 3 y ) = 2 ( 4 ) 4 x − 6 y = 8
Subtract the equations Now subtract the modified first equation from the second equation:
( 4 x − y ) − ( 4 x − 6 y ) = 13 − 8 4 x − y − 4 x + 6 y = 5 5 y = 5
Solve for y Solve for y :
y = 5 5 y = 1
Solve for x Substitute the value of y back into the first original equation to solve for x :
2 x − 3 ( 1 ) = 4 2 x − 3 = 4 2 x = 4 + 3 2 x = 7 x = 2 7
State the solution Therefore, the solution to the system of equations is x = 2 7 and y = 1 .
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 sells two products, we can set up a system of equations to determine the number of units of each product that need to be sold to reach a certain profit target. Solving this system helps the company make informed decisions about production and sales strategies.
