\begin{aligned} y & =6 x-9 \\ 2 x-6 y & =-48\end{aligned}
Answer (2)
Substitute the first equation into the second equation.
Solve the resulting equation for x : x = 3 .
Substitute the value of x back into the first equation to find y : y = 9 .
The solution to the system of equations is ( 3 , 9 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = 6 x − 9
2 x − 6 y = − 48
Our goal is to find the values of x and y that satisfy both equations. We can use the substitution method to solve this system.
Substitution Substitute the first equation, y = 6 x − 9 , into the second equation:
2 x − 6 ( 6 x − 9 ) = − 48
Solve for x Now, we solve for x :
2 x − 36 x + 54 = − 48
− 34 x = − 48 − 54
− 34 x = − 102
x = − 34 − 102
x = 3
Solve for y Now that we have the value of x , we can substitute it back into the first equation to find the value of y :
y = 6 ( 3 ) − 9
y = 18 − 9
y = 9
State the solution So, the solution to the system of equations is x = 3 and y = 9 . We can write this as an ordered pair ( 3 , 9 ) .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow. For example, a company might use a system of equations to determine how many units of a product they need to sell to cover their costs and start making a profit. Understanding how to solve systems of equations is a valuable skill in various fields.
To solve the given system of equations, we first substitute one equation into the other and simplify to find x and then y . The solution is ( 3 , 9 ) , meaning that when x = 3 , y = 9 .
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