Solve the given system of equations:
[tex]\begin{array}{l}
2 y=-x+9 \\
3 x-6 y=-15
\end{array}[/tex]
The solution to the system is ($\square$, $\square$).
Answer (2)
Rewrite the given equations: x + 2 y = 9 and x − 2 y = − 5 .
Eliminate y by adding the two equations: 2 x = 4 .
Solve for x : x = 2 .
Substitute x back into one of the equations to solve for y : y = 3.5 . The solution is ( 2 , 3.5 ) .
Explanation
Understanding the Problem 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. The equations are:
Stating the Equations 2 y = − x + 9 3 x − 6 y = − 15
Rewriting the First Equation First, let's rewrite the first equation to express x in terms of y or vice versa. We can rewrite the first equation as:
Simplified First Equation x + 2 y = 9
Rewriting the Second Equation Now, let's rewrite the second equation:
Stating the Second Equation 3 x − 6 y = − 15
Simplifying the Second Equation We can simplify the second equation by dividing all terms by 3:
Simplified Second Equation x − 2 y = − 5
System of Equations Now we have a system of two equations:
Stating the System x + 2 y = 9 x − 2 y = − 5
Elimination Method We can use the elimination method to solve this system. Add the two equations together to eliminate y :
Adding the Equations ( x + 2 y ) + ( x − 2 y ) = 9 + ( − 5 ) 2 x = 4
Solving for x Now, solve for x :
Value of x x = 2 4 = 2
Substituting x into the First Equation Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use the first equation:
Substituting x 2 + 2 y = 9
Isolating y Subtract 2 from both sides:
Simplifying 2 y = 9 − 2 = 7
Solving for y Now, solve for y :
Value of y y = 2 7 = 3.5
Final Answer So the solution to the system of equations is x = 2 and y = 3.5 . Therefore, the solution to the system is ( 2 , 3.5 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of products to maximize profit, or modeling supply and demand in economics. For instance, if a company sells two products, the system of equations can help determine the number of units of each product that need to be sold to cover the costs and achieve a certain profit target. By solving the system, the company can make informed decisions about pricing, production, and sales strategies. Let's say a company sells lemonade for $2 per cup and cookies for $1.50 each. They want to make a profit of $100, and they know they need to sell twice as many cups of lemonade as cookies. This scenario can be modeled as a system of equations, helping them determine the exact quantities to sell.
The solution to the system of equations is (2, 3.5). This was found by rewriting the equations, using the elimination method, and solving for both variables. The final answer shows the values of x and y that satisfy both equations.
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