What is the solution to the system of equations?
[tex]
\begin{array}{l}
y=\frac{2}{3} x+3 \\
x=-2
\end{array}
[/tex]
Answer (1)
Substitute the value of x from the second equation into the first equation: y = 3 2 ( − 2 ) + 3 .
Simplify the expression: y = − 3 4 + 3 .
Find a common denominator and add the fractions: y = − 3 4 + 3 9 = 3 5 .
The solution to the system of equations is ( − 2 , 3 5 ) .
Explanation
Understanding the problem We are given a system of two equations:
y = 3 2 x + 3
x = − 2
Our goal is to find the values of x and y that satisfy both equations.
Substituting the value of x Since we already know that x = − 2 , we can substitute this value into the first equation to solve for y :
y = 3 2 ( − 2 ) + 3
Simplifying the expression Now, let's simplify the expression to find the value of y :
y = − 3 4 + 3
To add these numbers, we need a common denominator. We can rewrite 3 as 3 9 :
y = − 3 4 + 3 9
Calculating the value of y Now we can add the fractions:
y = 3 − 4 + 9
y = 3 5
Stating the solution So, the solution to the system of equations is x = − 2 and y = 3 5 . Therefore, the solution as an ordered pair is ( − 2 , 3 5 ) .
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company's cost function is C ( x ) = 2 x + 100 and its revenue function is R ( x ) = 5 x , where x is the number of units, solving the system of equations y = 2 x + 100 and y = 5 x will give the break-even point, where costs equal revenue. This helps businesses make informed decisions about production and pricing.
