Solve the following linear system of equations:
[tex]\begin{array}{l}
3 x+y=21 \\
6 x-4 y=6
\end{array}[/tex]
x =
y =
Answer (2)
Multiply the first equation by 4: 12 x + 4 y = 84 .
Add the modified equation to the second equation: 18 x = 90 .
Solve for x : x = 5 .
Substitute x into the first equation and solve for y : y = 6 .
The solution is x = 5 , y = 6 .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, x and y :
Equation 1: 3 x + y = 21 Equation 2: 6 x − 4 y = 6
Our objective is to solve this system of equations to find the values of x and y . We can use either substitution or elimination method. Here, we will use the elimination method.
Modify Equation 1 Multiply the first equation by 4 to make the coefficients of y in both equations additive inverses:
4 ( 3 x + y ) = 4 ( 21 ) 12 x + 4 y = 84
Now we have:
Equation 1 (modified): 12 x + 4 y = 84 Equation 2: 6 x − 4 y = 6
Eliminate y and solve for x Add the modified Equation 1 to Equation 2 to eliminate y :
( 12 x + 4 y ) + ( 6 x − 4 y ) = 84 + 6 18 x = 90
Now, solve for x :
x = 18 90 = 5
Solve for y Substitute the value of x into the original Equation 1 to solve for y :
3 x + y = 21 3 ( 5 ) + y = 21 15 + y = 21 y = 21 − 15 = 6
State the solution Therefore, the solution to the system of equations is x = 5 and y = 6 .
Examples
Linear systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and analyzing electrical circuits. For example, a company might use a system of equations to determine how many units of two different products they need to sell to reach a specific revenue target, given the prices of the products and the costs involved. Solving such systems helps in making informed business decisions.
The solution to the given system of equations is x = 5 and y = 6 . We used the elimination method to find these values by modifying the equations and solving step-by-step. Finally, we substituted back to find the corresponding value of y .
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