Which graph represents the solution set to this system of equations?
[tex]$-x+2 y=6 \text { and } 4 x+y=3$[/tex]
Answer (2)
The solution to the system of equations − x + 2 y = 6 and 4 x + y = 3 is the point ( 0 , 3 ) . These lines intersect at this point on the graph. Therefore, the graph representing the solution set intersects at ( 0 , 3 ) .
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Use the elimination method to solve the system of equations.
Multiply the second equation by -2 and add it to the first equation to eliminate y.
Solve for x: x = 0 .
Substitute x = 0 into the first equation to solve for y: y = 3 .
The solution to the system of equations is ( 0 , 3 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
− x + 2 y = 6 4 x + y = 3
Our goal is to find the solution (x, y) to this system and identify the graph that represents it.
Eliminating y and Solving for x We can solve this system of equations using the substitution or elimination method. Let's use the elimination method. Multiply the second equation by -2 to eliminate y:
− x + 2 y = 6 − 8 x − 2 y = − 6
Add the two equations:
( − x − 8 x ) + ( 2 y − 2 y ) = ( 6 − 6 ) − 9 x = 0 x = 0
Solving for y Now that we have the value of x, we can substitute it into either of the original equations to solve for y. Let's use the first equation:
− x + 2 y = 6 − ( 0 ) + 2 y = 6 2 y = 6 y = 3
Finding the Intersection Point So the solution to the system of equations is (0, 3). This means the two lines intersect at the point (0, 3).
Final Answer The solution to the system of equations is x = 0 and y = 3 . Therefore, the graph that represents the solution set to this system of equations is the one where the two lines intersect at the point (0, 3).
Examples
Systems of equations are used in many 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 this case, solving the system of equations helps us find the point where two lines intersect, which can be useful in various scenarios like determining the location of a meeting point or analyzing the behavior of two related variables.
