Solve the following system of simultaneous equations:
[tex]
\begin{array}{r}
y-2 x=10 \\
2 x+5 y=26
\end{array}
[/tex]
Answer (2)
Solve the first equation for y : y = 2 x + 10 .
Substitute this expression for y into the second equation and solve for x : 2 x + 5 ( 2 x + 10 ) = 26 , which simplifies to x = − 2 .
Substitute the value of x back into the equation for y and solve for y : y = 2 ( − 2 ) + 10 , which gives y = 6 .
The solution to the system of equations is x = − 2 , y = 6 .
Explanation
Analyze 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 simultaneously. The equations are:
y − 2 x = 10
2 x + 5 y = 26
Solve for y in equation (1) We can use the substitution method to solve this system. First, solve equation (1) for y :
y = 2 x + 10
Substitute into equation (2) Now, substitute this expression for y into equation (2):
2 x + 5 ( 2 x + 10 ) = 26
Solve for x Simplify and solve for x :
2 x + 10 x + 50 = 26
12 x + 50 = 26
12 x = 26 − 50
12 x = − 24
x = 12 − 24
x = − 2
Substitute x into equation for y Substitute the value of x back into the equation for y :
y = 2 ( − 2 ) + 10
Solve for y Solve for y :
y = − 4 + 10
y = 6
State the solution Therefore, the solution to the system of equations is x = − 2 and y = 6 .
Examples
Simultaneous equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. For example, a company might use simultaneous equations to determine the number of units they need to sell to cover their costs and start making a profit. By setting up equations that represent the company's revenue and expenses, they can solve for the break-even point, which is the point where revenue equals expenses. This helps them make informed decisions about pricing, production, and marketing strategies.
The solution to the system of equations is x = − 2 and y = 6 . We used the substitution method by first solving for y in terms of x in the first equation, and then substituting this into the second equation to find the values of both variables. The steps led us to the final values of the variables that satisfy both equations simultaneously.
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