Question 1

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
y = - 3 x + 14 \
y = - \frac { 4 } { 3 } x + \frac { 22 } { 3 }\
(4, 2)
\end{array}$$
A)solution
B)not a solution

Accepted Answer

Substitute x = 4 into both equations:  1st equation: y = -3(4) + 14 = 2  2nd equation: y = -4/3(4) + 22/3 = 2  Both equations are satisfied with (4, 2).

Question 2

Find the solution to the system of equations by substitution.
-$$\begin{array} { c } 
x + 6 y = - 45 \
- 6 x + 5 y = - 17
\end{array}$$
A)(3, -6)
B)(-3, -7)
C)(-4, -6)
D)no solution

Accepted Answer

First, solve the first equation for $x$: $x = -45 - 6y$. Then substitute this expression for $x$ in the second equation: $-6(-45 - 6y) + 5y = -17$. Simplifying gives $270 + 36y + 5y = -17$, which simplifies to $41y = -287$, so $y = -7$. Substitute $y = -7$ back into the first equation to find $x$: $x + 6(-7) = -45$, which simplifies to $x - 42 = -45$, so $x = -3$. Therefore, the solution is $(-3, -7)$.

Question 3

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state.
-$$\begin{array} { l } 
y = - 9 x \
y = 9 x - 18
\end{array}$$
A)(1, 18)
B)(-1, 9)
C)(1, -9)
D)dependent

Accepted Answer

The solution to the system of equations is the point where the two lines intersect. Graphically, the lines y = -9x and y = 9x - 18 intersect at the point (1, -9).

Question 4

Find the solution to the system of equations by substitution.
-$$\begin{array} { l } 
x + 4 y = 7 \
- 3 x - 12 y = - 21
\end{array}$$
A)(7, 0)
B)(0, 0)
C)infinite number of solutions
D)no solution

Accepted Answer

The second equation is -3 times the first equation, indicating that both equations represent the same line. Therefore, there are an infinite number of solutions.

Question 5

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
y = - x - 1 \
x - y = - 7 \
( 4,3 )
\end{array}$$
A)solution
B)not a solution

Accepted Answer

To determine if the ordered pair (4,3) is a solution, substitute x=4 and y=3 into both equations. For the first equation, y = -x - 1 becomes 3 = -(4) - 1, which simplifies to 3 = -5, which is false. Therefore, the ordered pair (4,3) does not satisfy the system of equations.

Question 6

Find the solution to the system of equations by substitution.
-$$\begin{array} { l } 
x + y = - 6 \
x - y = 15
\end{array}$$
A)(4.5, -10.5)
B)(6, -10.5)
C)(4.5, 10.5)
D)(6, 4.5)

Accepted Answer

Solving the first equation for $x$, we get $x = -6 - y$. Substituting this into the second equation gives $(-6 - y) - y = 15$, which simplifies to $-6 - 2y = 15$, leading to $2y = -21$, so $y = -10.5$. Substituting $y = -10.5$ back into $x = -6 - y$ gives $x = 4.5$. Therefore, the solution is $(4.5, -10.5)$.

Question 7

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state.
-$$\begin{array} { c } 
4 x - 5 y = 4 \
15 y = 12 x - 8
\end{array}$$
A)(4, 4)
B) $\left( 2 , - \frac { 5 } { 2 } \right)$
C)dependent
D)inconsistent

Accepted Answer

The two equations represent parallel lines, as they have the same slope but different y-intercepts, making the system inconsistent with no solutions.

Question 8

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
y = - x - 1 \
x - y = - 9 \
( - 5,4 )
\end{array}$$
A)solution
B)not a solution

Accepted Answer

The ordered pair (-5, 4) satisfies both equations. For the first equation, substituting x = -5 gives y = -(-5) - 1 = 5 - 1 = 4, which matches the given y-value. For the second equation, substituting x = -5 and y = 4 gives -5 - 4 = -9, which matches the equation. Therefore, the ordered pair is a solution to the system.

Question 9

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
4 x + 2 y + z = - 14 \
2 x - 2 y - z = 8 \
5 x + y + 5 z = - 28 \
( - 4 , - 3 , - 1 )
\end{array}$$
A)solution
B)not a solution

Accepted Answer

The answer of Determine whether the ordered pair or ordered...

Question 10

Write each equation in slope-intercept form. Without graphing the equations, state whether the system of equations is
consistent, inconsistent, or dependent. Also indicate whether the system has exactly one solution, no solution, or an
infinite number of solutions.
-$$\begin{array} { l } 
x + 7 y = - 2 \
- 4 x - 28 y = 8
\end{array}$$
A)consistent
B)inconsistent
C)dependent one solution no solution infinite number of solutions

Accepted Answer

dependent, infinite number of solutions.The second equation is a multiple of the first (multiplying the first equation by -4 gives the second), indicating the lines are coincident (the same line), so the system is dependent with an infinite number of solutions.

Question 11

Write each equation in slope-intercept form. Without graphing the equations, state whether the system of equations is
consistent, inconsistent, or dependent. Also indicate whether the system has exactly one solution, no solution, or an
infinite number of solutions.
-$$\begin{array} { l } 
2 x - 6 y = 5 \
- 8 x + 24 y = - 15
\end{array}$$
A)dependent
B)consistent
C)inconsistent infinite number of solutions one solution no solution

Accepted Answer

The equations are inconsistent because when simplified, they represent parallel lines with no intersection, hence no solution.

Question 12

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state.
-$$y = - \frac { 7 } { 5 } x - \frac { 8 } { 5 }$$ $$y = - \frac { 7 } { 9 } x - \frac { 20 } { 9 }$$
A)(1, -3)
B)(-7, -3)
C)(7, 3)
D)(-1, 3)

Accepted Answer

The slopes of the two lines are different (-7/5 and -7/9), so they will intersect at exactly one point. By graphing or solving the system algebraically, you can find that the solution is (1, -3).

Question 13

Write each equation in slope-intercept form. Without graphing the equations, state whether the system of equations is
consistent, inconsistent, or dependent. Also indicate whether the system has exactly one solution, no solution, or an
infinite number of solutions.
-$$\begin{array} { c } 
x + 3 y = 3 \
8 x - 5 y = - 5
\end{array}$$
A)inconsistent
B)dependent
C)consistent no solution infinite number of solutions one solution

Accepted Answer

The equations in slope-intercept form are $y = -\frac{1}{3}x + 1$ and $y = \frac{8}{5}x - 1$. Since the slopes are different ($-\frac{1}{3} \neq \frac{8}{5}$), the lines are not parallel and not the same line, indicating they will intersect at exactly one point. Therefore, the system is consistent and has exactly one solution.

Question 14

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
y = - 3 x - 7 \
y = - \frac { 4 } { 3 } x - \frac { 1 } { 3 } \
( - 4 , - 5 )
\end{array}$$
A)not a solution
B)solution

Accepted Answer

To determine if the ordered pair $(-4, -5)$ is a solution to the system of equations, we substitute $x = -4$ and $y = -5$ into each equation and check for equality. For the first equation, $y = -3(-4) - 7 = 12 - 7 = 5$, which does not equal -5. For the second equation, $y = -\frac{4}{3}(-4) - \frac{1}{3} = \frac{16}{3} - \frac{1}{3} = \frac{15}{3} = 5$, which also does not equal -5. Since the ordered pair does not satisfy either equation, it is not a solution to the system.

Question 15

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state.
-$$\begin{array} { l } 
y = - x - 1 \
y = x - 16
\end{array}$$
A)(1, -8.5)
B)(7.5, 8.5)
C)(7.5, -8.5)
D)inconsistent

Accepted Answer

The solution to the system of equations is the point where the two lines intersect. By solving the equations $y = -x - 1$ and $y = x - 16$ graphically or algebraically, we find that they intersect at the point (7.5, -8.5).

Question 16

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state.
-$$\begin{array} { l } 
y = - 3 x - 1 \
- 6 x - 2 y = 2
\end{array}$$
A)(0, -1)
B)(0, 0)
C)dependent
D)inconsistent

Accepted Answer

The second equation can be simplified to $y = -3x - 1$, which is identical to the first equation, indicating the system is dependent (infinite solutions).

Question 17

Find the solution to the system of equations by substitution.
-$$\begin{array} { c } 
7 x + y = 0 \
- 7 x + y = - 14
\end{array}$$
A)(1, 14)
B)(-1, 7)
C)(1, -7)
D)(-1, -7)

Accepted Answer

From the first equation, we can express $y$ in terms of $x$: $y = -7x$. Substituting this expression for $y$ in the second equation gives $-7x + (-7x) = -14$, which simplifies to $-14x = -14$. Solving for $x$ gives $x = 1$. Substituting $x = 1$ back into the expression for $y$ gives $y = -7(1) = -7$. Therefore, the solution is $x = 1$, $y = -7$, which corresponds to choice C.

Question 18

Find the solution to the system of equations by substitution.
-$$\begin{array} { l } 
y = - 4 x + 26 \
y = - \frac { 1 } { 2 } x + 5
\end{array}$$
A)(6, 2)
B)(2, 6)
C)infinite number of solutions
D)no solution

Accepted Answer

To find the solution, set the equations equal to each other: $-4x + 26 = -\frac{1}{2}x + 5$. Solving for $x$, we get $x = 6$. Substitute $x = 6$ into either equation to find $y = 2$. Thus, the solution is (6, 2).

Question 19

Find the solution to the system of equations by substitution.
-$$\begin{array} { l } 
x + 2 y = 2 \
7 x - 6 y = - 6
\end{array}$$
A)(1, 0)
B)(0, 0)
C)(1, 1)
D)(0, 1)

Accepted Answer

The answer of Find the solution to the system of...

Question 20

Determine whether the ordered pair or ordered triple satisfies the system of linear equations.
-$$\begin{array} { l } 
4 x + 4 y + z = 15 \
4 x - 5 y - z = - 36 \
4 x + y + 3 z = 6 \
( - 2,5,3 )
\end{array}$$
A)solution
B)not a solution

Accepted Answer

Substituting $(-2, 5, 3)$ into each equation:  1. $4(-2) + 4(5) + 3 = 15$ simplifies to $-8 + 20 + 3 = 15$, which is true.  2. $4(-2) - 5(5) - 3 = -36$ simplifies to $-8 - 25 - 3 = -36$, which is true.  3. $4(-2) + 5 + 3(3) = 6$ simplifies to $-8 + 5 + 9 = 6$, which is true.  Thus, $(-2, 5, 3)$ satisfies all equations.

Quiz 4: Systems of Equations and Inequalities

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } y = - 3 x + 14 \\y = - \frac { 4 } { 3 } x + \frac { 22 } { 3 }\\(4, 2) \end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { c } x + 6 y = - 45 \\- 6 x + 5 y = - 17\end{array}$

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - $\begin{array} { l } y = - 9 x \\y = 9 x - 18\end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { l } x + 4 y = 7 \\- 3 x - 12 y = - 21\end{array}$

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } y = - x - 1 \\x - y = - 7 \\( 4,3 ) \end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { l } x + y = - 6 \\x - y = 15\end{array}$

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - $\begin{array} { c } 4 x - 5 y = 4 \\15 y = 12 x - 8\end{array}$

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } y = - x - 1 \\x - y = - 9 \\( - 5,4 ) \end{array}$

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } 4 x + 2 y + z = - 14 \\2 x - 2 y - z = 8 \\5 x + y + 5 z = - 28 \\( - 4 , - 3 , - 1 ) \end{array}$

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - $y = - \frac { 7 } { 5 } x - \frac { 8 } { 5 }$ $y = - \frac { 7 } { 9 } x - \frac { 20 } { 9 }$

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } y = - 3 x - 7 \\y = - \frac { 4 } { 3 } x - \frac { 1 } { 3 } \\( - 4 , - 5 ) \end{array}$

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - $\begin{array} { l } y = - x - 1 \\y = x - 16\end{array}$

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - $\begin{array} { l } y = - 3 x - 1 \\- 6 x - 2 y = 2\end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { c } 7 x + y = 0 \\- 7 x + y = - 14\end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { l } y = - 4 x + 26 \\y = - \frac { 1 } { 2 } x + 5\end{array}$

Find the solution to the system of equations by substitution. - $\begin{array} { l } x + 2 y = 2 \\7 x - 6 y = - 6\end{array}$

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - $\begin{array} { l } 4 x + 4 y + z = 15 \\4 x - 5 y - z = - 36 \\4 x + y + 3 z = 6 \\( - 2,5,3 ) \end{array}$

Quiz 4: Systems of Equations and Inequalities

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - y=−3x+14y=−43x+223(4,2) \begin{array} { l } y = - 3 x + 14 \\y = - \frac { 4 } { 3 } x + \frac { 22 } { 3 }\\(4, 2) \end{array}y=−3x+14y=−34​x+322​(4,2) ​

Find the solution to the system of equations by substitution. - x+6y=−45−6x+5y=−17\begin{array} { c } x + 6 y = - 45 \\- 6 x + 5 y = - 17\end{array}x+6y=−45−6x+5y=−17​

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - y=−9xy=9x−18\begin{array} { l } y = - 9 x \\y = 9 x - 18\end{array}y=−9xy=9x−18​

Find the solution to the system of equations by substitution. - x+4y=7−3x−12y=−21\begin{array} { l } x + 4 y = 7 \\- 3 x - 12 y = - 21\end{array}x+4y=7−3x−12y=−21​

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - y=−x−1x−y=−7(4,3) \begin{array} { l } y = - x - 1 \\x - y = - 7 \\( 4,3 ) \end{array}y=−x−1x−y=−7(4,3) ​

Find the solution to the system of equations by substitution. - x+y=−6x−y=15\begin{array} { l } x + y = - 6 \\x - y = 15\end{array}x+y=−6x−y=15​

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - 4x−5y=415y=12x−8\begin{array} { c } 4 x - 5 y = 4 \\15 y = 12 x - 8\end{array}4x−5y=415y=12x−8​

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - y=−x−1x−y=−9(−5,4) \begin{array} { l } y = - x - 1 \\x - y = - 9 \\( - 5,4 ) \end{array}y=−x−1x−y=−9(−5,4) ​

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - y=−75x−85y = - \frac { 7 } { 5 } x - \frac { 8 } { 5 }y=−57​x−58​ y=−79x−209y = - \frac { 7 } { 9 } x - \frac { 20 } { 9 }y=−97​x−920​

Determine whether the ordered pair or ordered triple satisfies the system of linear equations. - y=−3x−7y=−43x−13(−4,−5) \begin{array} { l } y = - 3 x - 7 \\y = - \frac { 4 } { 3 } x - \frac { 1 } { 3 } \\( - 4 , - 5 ) \end{array}y=−3x−7y=−34​x−31​(−4,−5) ​

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - y=−x−1y=x−16\begin{array} { l } y = - x - 1 \\y = x - 16\end{array}y=−x−1y=x−16​

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. - y=−3x−1−6x−2y=2\begin{array} { l } y = - 3 x - 1 \\- 6 x - 2 y = 2\end{array}y=−3x−1−6x−2y=2​

Find the solution to the system of equations by substitution. - 7x+y=0−7x+y=−14\begin{array} { c } 7 x + y = 0 \\- 7 x + y = - 14\end{array}7x+y=0−7x+y=−14​

Find the solution to the system of equations by substitution. - y=−4x+26y=−12x+5\begin{array} { l } y = - 4 x + 26 \\y = - \frac { 1 } { 2 } x + 5\end{array}y=−4x+26y=−21​x+5​

Find the solution to the system of equations by substitution. - x+2y=27x−6y=−6\begin{array} { l } x + 2 y = 2 \\7 x - 6 y = - 6\end{array}x+2y=27x−6y=−6​