Explain what happens when you try to solve this linear equation using an elimination strategy.
What does this tell you about the graphs of these equations?
-8x + 20y = -40
24x - 60y = 120
-8x + 20y = -40 ----> 2x - 5y = 10
24x - 60y = 120 ----- > 2x - 5y = 12
Ahhh, I see two parallel lines, thus no solution
will it work if i write :
they have the same slope and are identical. Both have the slope of 2/5?
But they are not "identical" , their constants are different.
You have 2 different parallel lines
(think of railway tracks)
When you try to solve a system of linear equations using the elimination strategy, the goal is to eliminate one variable by adding or subtracting the given equations. This allows you to solve for the remaining variable.
Let's analyze the given equations:
-8x + 20y = -40 (Equation 1)
24x - 60y = 120 (Equation 2)
To eliminate one variable, we need to manipulate these equations in a way that the coefficients of either the x or y terms are opposites.
Step 1: Multiply Equation 1 by 3 and Equation 2 by -1 to make the coefficients of x in both equations the same:
-24x + 60y = -120 (Equation 1')
-24x + 60y = 120 (Equation 2')
Step 2: Now that the coefficients of x are opposites, we can add the two equations together to eliminate the x term:
(-24x + 60y) + (-24x + 60y) = (-120) + 120
-48x + 120y = 0
At this point, we only have one equation in terms of the remaining variable, y. We can solve for y by isolating it:
-48x + 120y = 0
120y = 48x
y = (48/120)x
y = (2/5)x
Now, let's analyze what this elimination strategy tells us about the graphs of these equations. By eliminating one variable, we have reduced the original system of equations to a single equation: y = (2/5)x. This equation represents a line in the xy-plane.
Therefore, the two original equations represent two lines on the graph. The solution to the system of equations will be the point of intersection between these two lines. In this case, since we have reduced the system to a single equation, the two lines are the same and will intersect at infinitely many points along this line.
Hence, the graphs of these equations are coincident or overlapping lines, indicating that they have infinitely many solutions.