Solve the system of linear equations by graphing.
[tex]
\begin{array}{l}
y=-\frac{5}{2} x-7 \\
x+2 y=4
\end{array}
[/tex]
What is the solution to the system of linear equations?
A. (-4.5, 4.25)
B. (0,-7)
C. (-1.7,-2.8)
D. (3, 0.5)
Answer (1)
Solve the second equation for x : x = 4 − 2 y .
Substitute the expression for x into the first equation and solve for y : y = 4.25 .
Substitute the value of y back into the equation for x and solve for x : x = − 4.5 .
The solution to the system of equations is ( − 4.5 , 4.25 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = − 2 5 x − 7
x + 2 y = 4
We need to find the solution to this system, which represents the point where the two lines intersect. We are given four possible solutions and we need to determine which one is correct.
Solving the System of Equations We can substitute each of the given points into both equations to see which point satisfies both equations. Alternatively, we can solve the system of equations using substitution or elimination. Let's solve the system algebraically. First, we can solve the second equation for x :
x = 4 − 2 y
Now, substitute this expression for x into the first equation:
y = − 2 5 ( 4 − 2 y ) − 7
Simplify and solve for y :
y = − 10 + 5 y − 7
y − 5 y = − 17
− 4 y = − 17
y = 4 17 = 4.25
Now, substitute y = 4.25 back into the equation x = 4 − 2 y :
x = 4 − 2 ( 4.25 )
x = 4 − 8.5
x = − 4.5
So the solution to the system of equations is x = − 4.5 and y = 4.25 .
Finding the Correct Option The solution we found is ( − 4.5 , 4.25 ) . Now we check the given options to see if this solution is among them.
Final Answer The solution to the system of linear equations is ( − 4.5 , 4.25 ) .
Examples
Systems of linear 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 per unit, and they sell each unit 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. Another example is in network flow problems, where we want to determine the optimal flow of goods or information through a network, subject to certain constraints. These problems can often be modeled as systems of linear equations.
