Question

match each system on the left with all words that describe the system on the right. choices on the right can be used more than once.
$y = 2x + 3$
$x + y = -3$
$3y = 9x - 6$
$2y - 6x = 4$
$y = -\frac{1}{2}x + 2$
$x + 2y = 4$

inconsistent
consistent
independent
dependent

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clear

Explanation:

To solve this, we analyze each system of linear equations by converting them to slope - intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y - intercept) and then determine the relationship between the lines (parallel, intersecting, coinciding) to classify the system.

System 1: $\boldsymbol{y = 2x + 3}$ and $\boldsymbol{x + y=-3}$

1. First, rewrite the second equation in slope - intercept form:

• Start with $x + y=-3$.
• Subtract $x$ from both sides: $y=-x - 3$.
• The first equation $y = 2x+3$ has a slope $m_1 = 2$ and y - intercept $b_1 = 3$. The second equation $y=-x - 3$ has a slope $m_2=-1$ and y - intercept $b_2=-3$.
• Since the slopes are different ($m_1

eq m_2$), the two lines intersect at exactly one point. A system of linear equations with exactly one solution is consistent (because there is a solution) and the equations are independent (because they represent two different non - parallel lines).

System 2: $\boldsymbol{3y = 9x-6}$ and $\boldsymbol{2y-6x = 4}$

1. Rewrite both equations in slope - intercept form:

• For the first equation $3y = 9x-6$, divide both sides by 3: $y = 3x-2$. So, $m_1 = 3$ and $b_1=-2$.
• For the second equation $2y-6x = 4$, add $6x$ to both sides: $2y=6x + 4$. Then divide by 2: $y = 3x+2$. So, $m_2 = 3$ and $b_2 = 2$.
• Since the slopes are equal ($m_1=m_2$) but the y - intercepts are different ($b_1

eq b_2$), the lines are parallel. A system of parallel lines has no solution, so it is inconsistent (no solution exists) and the equations are independent (two different parallel lines).

System 3: $\boldsymbol{y=-\frac{1}{2}x + 2}$ and $\boldsymbol{x + 2y=4}$

1. Rewrite the second equation in slope - intercept form:

• Start with $x + 2y=4$. Subtract $x$ from both sides: $2y=-x + 4$. Divide by 2: $y=-\frac{1}{2}x + 2$.
• The first equation is $y=-\frac{1}{2}x + 2$ and the second equation, after rearrangement, is also $y=-\frac{1}{2}x + 2$.
• Since the two equations represent the same line (same slope and same y - intercept), the system has infinitely many solutions. A system with infinitely many solutions is consistent (solutions exist) and the equations are dependent (one equation is a multiple of the other, they represent the same line).

Final Matches:

• $y = 2x + 3$ and $x + y=-3$: consistent, independent
• $3y = 9x-6$ and $2y - 6x=4$: inconsistent, independent
• $y=-\frac{1}{2}x + 2$ and $x + 2y = 4$: consistent, dependent

Answer:

To solve this, we analyze each system of linear equations by converting them to slope - intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y - intercept) and then determine the relationship between the lines (parallel, intersecting, coinciding) to classify the system.

System 1: $\boldsymbol{y = 2x + 3}$ and $\boldsymbol{x + y=-3}$

1. First, rewrite the second equation in slope - intercept form:

• Start with $x + y=-3$.
• Subtract $x$ from both sides: $y=-x - 3$.
• The first equation $y = 2x+3$ has a slope $m_1 = 2$ and y - intercept $b_1 = 3$. The second equation $y=-x - 3$ has a slope $m_2=-1$ and y - intercept $b_2=-3$.
• Since the slopes are different ($m_1

eq m_2$), the two lines intersect at exactly one point. A system of linear equations with exactly one solution is consistent (because there is a solution) and the equations are independent (because they represent two different non - parallel lines).

System 2: $\boldsymbol{3y = 9x-6}$ and $\boldsymbol{2y-6x = 4}$

1. Rewrite both equations in slope - intercept form:

• For the first equation $3y = 9x-6$, divide both sides by 3: $y = 3x-2$. So, $m_1 = 3$ and $b_1=-2$.
• For the second equation $2y-6x = 4$, add $6x$ to both sides: $2y=6x + 4$. Then divide by 2: $y = 3x+2$. So, $m_2 = 3$ and $b_2 = 2$.
• Since the slopes are equal ($m_1=m_2$) but the y - intercepts are different ($b_1

eq b_2$), the lines are parallel. A system of parallel lines has no solution, so it is inconsistent (no solution exists) and the equations are independent (two different parallel lines).

System 3: $\boldsymbol{y=-\frac{1}{2}x + 2}$ and $\boldsymbol{x + 2y=4}$

1. Rewrite the second equation in slope - intercept form:

• Start with $x + 2y=4$. Subtract $x$ from both sides: $2y=-x + 4$. Divide by 2: $y=-\frac{1}{2}x + 2$.
• The first equation is $y=-\frac{1}{2}x + 2$ and the second equation, after rearrangement, is also $y=-\frac{1}{2}x + 2$.
• Since the two equations represent the same line (same slope and same y - intercept), the system has infinitely many solutions. A system with infinitely many solutions is consistent (solutions exist) and the equations are dependent (one equation is a multiple of the other, they represent the same line).

Final Matches:

• $y = 2x + 3$ and $x + y=-3$: consistent, independent
• $3y = 9x-6$ and $2y - 6x=4$: inconsistent, independent
• $y=-\frac{1}{2}x + 2$ and $x + 2y = 4$: consistent, dependent