What is the point of intersection when the system of equations below is graphed on the coordinate plane?
[tex]\left\{\begin{array}{l}
-x+y=4 \\
6 x+y=-3
\end{array}\right.[/tex]
A. $(1,-3)$
B. $(-1,3)$
C. $(1,3)$
D. $(-1,-3)$
Answer (2)
Use the elimination method to solve the system of equations.
Subtract the first equation from the second equation to eliminate y and solve for x : x = − 1 .
Substitute the value of x into the first equation to solve for y : y = 3 .
The point of intersection is ( − 1 , 3 ) .
Explanation
Understanding the Problem We are given a system of two linear equations and asked to find the point of intersection when the equations are graphed. This point is the solution to the system of equations. We can solve this system using either the substitution or elimination method.
Elimination Method Let's use the elimination method. The given equations are: − x + y = 4 6 x + y = − 3 Subtract the first equation from the second equation to eliminate y :
( 6 x + y ) − ( − x + y ) = − 3 − 4 6 x + y + x − y = − 7 7 x = − 7 Divide both sides by 7: x = − 1
Solving for y Now substitute the value of x = − 1 into the first equation to solve for y :
− ( − 1 ) + y = 4 1 + y = 4 Subtract 1 from both sides: y = 4 − 1 y = 3
Point of Intersection The solution to the system of equations is x = − 1 and y = 3 . Therefore, the point of intersection is ( − 1 , 3 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point in business, calculating the optimal mix of ingredients in manufacturing, and modeling supply and demand in economics. For example, a company might use a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve systems of equations is a fundamental skill in many fields.
The point of intersection for the given system of equations is found to be (-1, 3). This is determined via the elimination method. Therefore, the answer is option B: (-1, 3).
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