Question

student id#: ______________
reasoning- algebra crt systems problem
constructed response task
part a
a system of two linear equations is given below:
$x + y = 6$
$y = 3x + 2$
will this system always have a unique solution? explain why or why not.

part b
a system of two linear equations is given below:
$6x - 3y = 3$
$2x - y = 4$
will this system always have a unique solution? explain why or why not.

part c
a system of two linear equations is given below:
$y = 4 - x$
$2y + 2x = 8$
will this system always have a unique solution? explain why or why not.

Explanation:

Part A

Step1: Analyze slopes of equations

The first equation \(x + y = 6\) can be rewritten as \(y=-x + 6\), so its slope \(m_1=-1\). The second equation \(y = 3x+2\) has slope \(m_2 = 3\).

Step2: Compare slopes

Since \(m_1
eq m_2\), the two lines are not parallel (and not coincident as slopes are different). For a system of two linear equations (two lines), if they are not parallel (and not coincident), they intersect at exactly one point. So this system has a unique solution, and since the slopes are fixed (the equations are fixed linear equations), it will always have a unique solution.

Step1: Simplify the first equation

The first equation \(6x-3y = 3\) can be divided by 3 on both sides: \(2x - y=1\). The second equation is \(2x - y = 4\).

Step2: Analyze the equations

Now we have two equations: \(2x - y=1\) and \(2x - y = 4\). If we rewrite them in slope - intercept form (\(y=mx + b\)), both will be \(y = 2x-1\) and \(y = 2x - 4\). The slopes of both lines are equal (\(m = 2\)) and the y - intercepts (\(-1\) and \(-4\)) are different. So the two lines are parallel and will never intersect.

Step1: Simplify the second equation

The second equation \(2y + 2x = 8\) can be divided by 2 on both sides: \(y + x=4\), which is the same as \(y=-x + 4\). The first equation is \(y = 4 - x\) (which is also \(y=-x + 4\)).

Step2: Analyze the equations

Both equations are the same line. So every point on the line \(y=-x + 4\) is a solution of the system. There are infinitely many solutions, not a unique solution.

Answer:

Yes, this system will always have a unique solution. The two linear equations \(x + y = 6\) (or \(y=-x + 6\)) and \(y = 3x+2\) have slopes \(- 1\) and \(3\) respectively. Since the slopes are not equal, the two lines are not parallel (and not coincident), so they must intersect at exactly one point, meaning the system has a unique solution. And as the equations are fixed linear equations, this unique - solution property holds always.

Part B