SOLUTION: do the following two lines intersect? answer yes or no, together with the point of intersection if any. 5x + 6y= -5.5 6x+1.5y = -8.5

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Question 871632: do the following two lines intersect? answer yes or no, together with the point of intersection if any.
5x + 6y= -5.5
6x+1.5y = -8.5
Found 2 solutions by ewatrrr, stanbon:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
(-3/2, 1/3)
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%285%2Ax%2B6%2Ay=-5.5%2C6%2Ax%2B1.5%2Ay=-8.5%29



First let A=%28matrix%282%2C2%2C5%2C6%2C6%2C1.5%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are -5.5 and -8.5 which are highlighted here:
system%285%2Ax%2B6%2Ay=highlight%28-5.5%29%2C6%2Ax%2B1.5%2Ay=highlight%28-8.5%29%29



These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=%285%29%281.5%29-%286%29%286%29=-28.5. Remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc. If you need help with calculating the determinant of any two by two matrices, then check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



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Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).


A%5Bx%5D=%28matrix%282%2C2%2Chighlight%28-5.5%29%2C6%2Chighlight%28-8.5%29%2C1.5%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-5.5%29%281.5%29-%286%29%28-8.5%29=42.75. Once again, remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2842.75%29%2F%28-28.5%29=-1.5



So the first solution is x=-1.5




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We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C5%2C6%2C6%2C1.5%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).


A%5Bx%5D=%28matrix%282%2C2%2C5%2Chighlight%28-5.5%29%2C6%2Chighlight%28-8.5%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%285%29%28-8.5%29-%28-5.5%29%286%29=-9.5.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-9.5%29%2F%28-28.5%29=0.333333333333333



So the second solution is y=0.333333333333333




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Final Answer:




So the solutions are x=-1.5 and y=0.333333333333333 giving the ordered pair (-1.5, 0.333333333333333)




Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
do the following two lines intersect? answer yes or no, together with the point of intersection if any.
5x + 6y= -5.5
6x+1.5y = -8.5
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MOdifiy for elimination:
5x + 6y = -5.5
24x + 6y = -34
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Subtract to solve for "x":
19x = -28.5
x = -28.5/19 = -1.5
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Solve for "y":
5x + 6y = -5.5
5((-1.5) + 6y = -5.5
6y = 2
y = 1/3
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Cheers,
Stan H.
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