Solve the following system by graphing and identify the point of intersection.
$\left\{\begin{array}{l}
-5 x+2 y=-4 \\
2 x-y=2
\end{array}\right.$
A. $(1,2)$
B. $(0,-2)$
C. $(2,1)$
D. $(-2,0)$
Answer (1)
Rewrite the equations in slope-intercept form: y = 2 5 x − 2 and y = 2 x − 2 .
Test the given points in both equations.
The point (0, -2) satisfies both equations.
The solution to the system of equations is ( 0 , − 2 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
{ − 5 x + 2 y = − 4 2 x − y = 2
Our goal is to find the point of intersection by graphing. We can also verify the solution by substituting the given points into the equations.
Rewriting Equations Let's rewrite each equation in slope-intercept form ( y = m x + b ).
For the first equation, − 5 x + 2 y = − 4 , we solve for y :
2 y = 5 x − 4
y = 2 5 x − 2
For the second equation, 2 x − y = 2 , we solve for y :
y = 2 x − 2
Testing the Points Now, let's test the given points to see which one satisfies both equations.
(1, 2):
Equation 1: − 5 ( 1 ) + 2 ( 2 ) = − 5 + 4 = − 1 = − 4 . So, (1, 2) is not a solution.
Equation 2: 2 ( 1 ) − 2 = 0 = 2 . So, (1, 2) is not a solution.
(0, -2):
Equation 1: − 5 ( 0 ) + 2 ( − 2 ) = 0 − 4 = − 4 . So, (0, -2) satisfies the first equation.
Equation 2: 2 ( 0 ) − ( − 2 ) = 0 + 2 = 2 . So, (0, -2) satisfies the second equation.
Therefore, (0, -2) is a solution.
(2, 1):
Equation 1: − 5 ( 2 ) + 2 ( 1 ) = − 10 + 2 = − 8 = − 4 . So, (2, 1) is not a solution.
Equation 2: 2 ( 2 ) − 1 = 4 − 1 = 3 = 2 . So, (2, 1) is not a solution.
(-2, 0):
Equation 1: − 5 ( − 2 ) + 2 ( 0 ) = 10 + 0 = 10 = − 4 . So, (-2, 0) is not a solution.
Equation 2: 2 ( − 2 ) − 0 = − 4 = 2 . So, (-2, 0) is not a solution.
Finding the Solution The point (0, -2) satisfies both equations. Therefore, it is the solution to the system of equations.
Examples
Systems of equations are useful in many real-world scenarios. For instance, imagine you're running a lemonade stand. You sell small cups for $2 and large cups for $3. If you made $200 in a day and sold 80 cups, you could set up a system of equations to find out how many small and large cups you sold. Solving this system helps you manage your inventory and pricing strategies effectively.
