Solve the simultaneous equations
[tex]
\begin{array}{l}
6 x+5 y=13 \\
2 x+3 y=3
\end{array}
[/tex]
Answer (1)
Multiply the second equation by 3: 6 x + 9 y = 9 .
Subtract the modified second equation from the first: − 4 y = 4 .
Solve for y : y = − 1 .
Substitute y = − 1 into the second equation and solve for x : x = 3 . The solution is x = 3 , y = − 1 .
Explanation
Analyze 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. The equations are:
Equation 1: 6 x + 5 y = 13 Equation 2: 2 x + 3 y = 3
Eliminate x To solve this system, we can use the method of elimination. We want to eliminate one of the variables by making the coefficients of that variable the same in both equations. Let's eliminate x . We can multiply the second equation by 3 to make the coefficient of x equal to 6, which is the coefficient of x in the first equation.
Multiply Equation 2 by 3: 3 ( 2 x + 3 y ) = 3 ( 3 ) which simplifies to 6 x + 9 y = 9 .
Subtract the equations Now we have:
Equation 1: 6 x + 5 y = 13 Modified Equation 2: 6 x + 9 y = 9
Subtract the modified second equation from the first equation to eliminate x : ( 6 x + 5 y ) − ( 6 x + 9 y ) = 13 − 9 which simplifies to − 4 y = 4 .
Solve for y Solve for y : y = − 4 4 = − 1 .
Substitute y into equation 2 Now that we have the value of y , we can substitute it back into either of the original equations to solve for x . Let's use the second equation: 2 x + 3 y = 3 .
Substitute y = − 1 into the second equation: 2 x + 3 ( − 1 ) = 3 which simplifies to 2 x − 3 = 3 .
Solve for x Solve for x : 2 x = 6 , so x = 2 6 = 3 .
State the solution Therefore, the solution to the system of equations is x = 3 and y = − 1 .
Examples
Systems of equations are used in various real-world applications. For example, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. They are also used in engineering to analyze circuits and structures, and in computer graphics to perform transformations and solve for intersections.
