Solve the system of linear equations:
[tex]\begin{array}{l}
x-y=2 \
2 x+6 y=36 \
x=\square \
y=\square\\
\end{array}[/tex]
Answer (2)
Express x in terms of y using the first equation: x = y + 2 .
Substitute this expression into the second equation and solve for y : 2 ( y + 2 ) + 6 y = 36 ⇒ y = 4 .
Substitute the value of y back into the expression for x : x = 4 + 2 ⇒ x = 6 .
The solution to the system of equations is x = 6 , y = 4 .
Explanation
Problem Analysis 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:
x − y = 2
2 x + 6 y = 36
Expressing x in terms of y We can solve this system of equations using either substitution or elimination. Let's use the substitution method.
From the first equation, we can express x in terms of y :
x = y + 2
Solving for y Now, substitute this expression for x into the second equation:
2 ( y + 2 ) + 6 y = 36
Expand and simplify:
2 y + 4 + 6 y = 36
Combine like terms:
8 y + 4 = 36
Subtract 4 from both sides:
8 y = 32
Divide by 8:
y = 4
Solving for x Now that we have the value of y , we can substitute it back into the expression for x :
x = y + 2
x = 4 + 2
x = 6
Final Answer Therefore, the solution to the system of equations is x = 6 and y = 4 .
We can check our solution by substituting these values back into the original equations:
6 − 4 = 2 (True)
2 ( 6 ) + 6 ( 4 ) = 12 + 24 = 36 (True)
Both equations are satisfied, so our solution is correct.
Examples
Systems of linear 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 in producing them. Solving such systems helps in making informed decisions and optimizing outcomes.
To solve the system of equations, we express x in terms of y using the first equation, substitute into the second equation, and find y . The values we find are x = 6 and y = 4 , and verification shows both original equations are satisfied. Thus, the solution is confirmed as correct.
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