Solving Linear Equations - Elimination
Write whether each of these systems of equations has a common solution or not. Explain
y = -x + 2 Y = 2x + 1
If you show me one and explain it, then I can probably do the rest...Thanks
if both of your equations are expressed as y = ..., then they are easy to solve.
You simply equate the right sides, that is,
-x+2=2x+1
-3x = -1
x = 1/3
sub that back into either one of the two original equations to get the y
I does not matter which one, you will get the same answer,
so in the first
y = -(1/3) + 2 = 5/3
what you are doing is finding the intersection point,(the common solution), of your two lines.
It is (1/3,5/3)
To determine whether the given system of equations has a common solution or not, we need to compare the slopes and y-intercepts of the two equations.
The given system of equations is:
y = -x + 2 (equation 1)
y = 2x + 1 (equation 2)
First, let's compare the slopes of the two equations. The slope of equation 1 is -1, and the slope of equation 2 is 2.
If the slopes of the two lines are equal, then the system of equations has infinitely many common solutions. If the slopes are not equal, we can determine the solution by finding the point of intersection of the two lines.
In this case, the slopes are not equal since -1 ≠ 2. So, the system of equations has a unique solution, which is the point of intersection of the two lines.
Next, let's find the point of intersection by solving the system of equations using the method of elimination:
We can manipulate equation 1 by multiplying it by -2 so that the coefficients of y in both equations are equal:
-2(y = -x + 2)
-2y = 2x - 4
Now, we can add the modified equation 1 to equation 2 to eliminate the x variable:
-2y + y = 2x - 4 + 2x + 1
-y = 4x - 3
To isolate the variable y, multiply the entire equation by -1:
y = -4x + 3
Now we have a new equation for y. Let's compare it to the original equation 2:
y = 2x + 1 (equation 2)
y = -4x + 3 (equation 3)
By comparing equation 2 and equation 3, we can see that the slopes and y-intercepts are not equal. Therefore, the system of equations does not have a common solution.
In conclusion, the given system of equations, y = -x + 2 and y = 2x + 1, does not have a common solution.