$(3x - 2y, 7) = (11, x + y)
Answer (1)
Set up the equations 3 x − 2 y = 11 and x + y = 7 from the given ordered pair equation.
Express y in terms of x using the second equation: y = 7 − x .
Substitute this expression into the first equation and solve for x , obtaining x = 5 .
Substitute the value of x back into the equation for y to find y = 2 . The solution is x = 5 , y = 2 .
Explanation
Setting up the Equations We are given the equation ( 3 x − 2 y , 7 ) = ( 11 , x + y ) . This means that the corresponding components of the ordered pairs must be equal. Therefore, we have two equations:
The System of Equations
3 x − 2 y = 11
x + y = 7
Expressing y in terms of x We can solve this system of equations using substitution or elimination. Let's use substitution. From equation (2), we can express y in terms of x :
y = 7 − x
Substitution Now, substitute this expression for y into equation (1):
3 x − 2 ( 7 − x ) = 11
Solving for x Simplify and solve for x :
3 x − 14 + 2 x = 11
5 x = 25
x = 5
Solving for y Now that we have the value of x , we can find the value of y using the equation y = 7 − x :
y = 7 − 5
y = 2
The Solution Therefore, the solution to the system of equations is x = 5 and y = 2 .
Verification We can check our solution by substituting these values back into the original equations:
3 x − 2 y = 3 ( 5 ) − 2 ( 2 ) = 15 − 4 = 11 (Correct)
x + y = 5 + 2 = 7 (Correct)
Examples
Systems of equations are used extensively in real-world applications, such as in economics to model supply and demand curves, in physics to solve for forces and motion, and in computer graphics to perform transformations. For instance, if you're trying to determine the optimal price and quantity of a product to maximize profit, you might set up a system of equations representing cost and revenue. Solving this system will give you the equilibrium point where supply equals demand, helping you make informed business decisions.
