Type the correct answer in each box. If necessary, use / for the fraction bar(s).
Find the solution for this system of equations.
[tex]\begin{aligned}
2 x-3 y & =2 \\
x & =6 y-5
\end{aligned}[/tex]
x= [ ]
y= [ ]
Answer (2)
Substitute x = 6 y − 5 into the first equation.
Simplify the equation and solve for y : 2 ( 6 y − 5 ) − 3 y = 2 ⇒ 9 y = 12 ⇒ y = 3 4 .
Substitute y = 3 4 back into x = 6 y − 5 and solve for x : x = 6 ( 3 4 ) − 5 = 3 .
The solution is x = 3 and y = 3 4 , so the final answer is x = 3 , y = 3 4 .
Explanation
Analyze the problem We are given a system of two equations with two unknowns, x and y . Our goal is to find the values of x and y that satisfy both equations.
Equation 1: 2 x − 3 y = 2 Equation 2: x = 6 y − 5
Substitution We will use the substitution method to solve this system. Substitute the expression for x from Equation 2 into Equation 1:
2 ( 6 y − 5 ) − 3 y = 2
Solve for y Now, simplify and solve for y :
12 y − 10 − 3 y = 2
9 y − 10 = 2
9 y = 12
y = 9 12 = 3 4
Solve for x Substitute the value of y back into Equation 2 to find the value of x :
x = 6 ( 3 4 ) − 5
x = 3 24 − 5
x = 8 − 5
x = 3
State the solution Therefore, the solution to the system of equations is x = 3 and y = 3 4 .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell a product at a certain price, we can set up a system of equations to find the number of units they need to sell to cover their costs and start making a profit. This involves solving for the quantity and price where the cost and revenue equations intersect, providing valuable insights for business planning and decision-making.
The solution to the system of equations is x = 3 and y = 3 4 . By substituting x in terms of y and simplifying, we found the values for both variables. This method allows us to efficiently solve systems of equations.
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