The first equation from the previous system of equations is graphed. Graph the second equation to find the solution of the system of equations.
[tex]
\begin{array}{l}
y=-2 x+4 \\
y=-\frac{1}{3} x-1
\end{array}
[/tex]
What is the solution to the system?
[tex]\square[/tex]
Answer (1)
Set the two equations equal to each other: − 2 x + 4 = − 3 1 x − 1 .
Solve for x : x = 3 .
Substitute the value of x back into one of the equations to solve for y : y = − 2 ( 3 ) + 4 = − 2 .
The solution to the system of equations is ( 3 , − 2 ) , which represents the intersection point of the two lines: ( 3 , − 2 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = − 2 x + 4 y = − 3 1 x − 1
The first equation is already graphed. We need to graph the second equation and find the solution to the system. The solution is the intersection point of the two lines.
Solving for x and y To find the solution to the system of equations, we need to find the point where the two lines intersect. We can do this by setting the two equations equal to each other and solving for x :
− 2 x + 4 = − 3 1 x − 1
Now, we solve for x :
− 2 x + 3 1 x = − 1 − 4 − 3 5 x = − 5 x = − 5 × − 5 3 x = 3 Now that we have the value of x , we can substitute it back into either equation to find the value of y . Let's use the first equation:
y = − 2 ( 3 ) + 4 y = − 6 + 4 y = − 2 So, the solution to the system of equations is ( 3 , − 2 ) .
Finding the Solution The solution to the system of equations is the point of intersection of the two lines, which we found to be ( 3 , − 2 ) .
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 example, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. Graphing these equations helps visualize the relationships and find the point where the company breaks even.
