$\begin{aligned} y & =-x \\ 10 x+2 y & =40\end{aligned}$
Answer (2)
Substitute y = − x into the second equation.
Simplify and solve for x : 10 x + 2 ( − x ) = 40 ⇒ 8 x = 40 ⇒ x = 5 .
Substitute x = 5 back into y = − x to find y = − 5 .
The solution to the system of equations is x = 5 , y = − 5 .
Explanation
Analyze the problem We are given a system of two linear equations:
y 10 x + 2 y = − x = 40
Our goal is to find the values of x and y that satisfy both equations.
Substitution We can use the substitution method to solve this system. Since the first equation is already solved for y , we can substitute − x for y in the second equation:
10 x + 2 ( − x ) = 40
Solve for x Now, we simplify the equation and solve for x :
10 x − 2 x = 40 8 x = 40 x = 8 40 x = 5
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 = − x y = − 5
Final Answer Therefore, the solution to the system of equations is x = 5 and y = − 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 ingredients in a recipe, or modeling traffic flow in a city. In this case, we solved a simple system of two linear equations, which can be visualized as two lines intersecting at a single point on a graph. The solution (x, y) represents the coordinates of that intersection point. Understanding how to solve systems of equations is a fundamental skill in mathematics and has practical applications in many fields.
To solve the system of equations, we substitute y = − x into the second equation, leading to x = 5 . After finding x , we compute y to get y = − 5 . The solution to the system is ( x , y ) = ( 5 , − 5 ) .
;
