Question 1

Answer true or false.
-If the two equations in a system of linear equations are added and the result is 8 = 0, the system has no solution.

Accepted Answer

This is because the equation 8 = 0 is not true and therefore there cannot be a solution to the system of equations. This means that the two equations are inconsistent and cannot be satisfied simultaneously, so there is no solution.

Question 2

Answer true or false.
-Although (7, 3)is not a solution of x + 3y = 2, it can still be a solution of the system $\left\{ \begin{array} { l } 
x + 3 y = 2 \
x + y = 4
\end{array} \right.$

Accepted Answer

The fact that (7, 3) is not a solution to the first equation does not automatically exclude it from being a solution to the entire system. The system of equations could have different constraints, and a point not satisfying one equation might still satisfy another equation or the combination of equations in the system. However, in this specific case, substituting (7, 3) into both equations of the system will show that it does not actually satisfy either equation, but the question's logic about the possibility remains true.

Question 3

Solve the system of equations by the substitution method.
-$\left\{ \begin{array} { l } 
4 x - 3 y = - 15 \
y = x + 4
\end{array} \right.$

Accepted Answer

The answer of Solve the system of equations by the...

Question 4

Solve the system by the substitution or the addition method.
-$\left\{ \begin{array} { l } 
3 x + y = 4 \
4 x + 3 y = - 3
\end{array} \right.$

Accepted Answer

To solve the system, we can use the substitution method. First, solve the first equation for $y$: $y = 4 - 3x$. Then substitute this expression for $y$ in the second equation: $4x + 3(4 - 3x) = -3$. Simplifying gives $4x + 12 - 9x = -3$, which simplifies further to $-5x = -15$, so $x = 3$. Substituting $x = 3$ back into the first equation gives $3(3) + y = 4$, so $9 + y = 4$, and $y = -5$. Therefore, the solution is $x = 3$, $y = -5$, which corresponds to choice A.

Question 5

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-In a system of linear equations in two variables, if the graphs of the equations are the same, the equations are_____________ equations.

Accepted Answer

If the graphs of the equations are the same, it means that they are representing the same line. This implies that there are infinitely many solutions that satisfy both equations, which means that the equations are dependent. If the graphs of the equations are the same, it means that the equations are dependent.

Question 6

Is the ordered pair a solution of the linear system?
-$\left\{ \begin{array} { l } 
5 x - 3 y = 21 \
4 x + 5 y = - 3
\end{array} ; \quad ( 3 , - 2 ) \right.$

Accepted Answer

Substituting $x = 3$ and $y = -2$ into both equations: For the first equation, $5(3) - 3(-2) = 15 + 6 = 21$, which matches the right side of the equation. For the second equation, $4(3) + 5(-2) = 12 - 10 = 2$, which does not match the right side of the equation (-3). My initial assessment was incorrect; the correct evaluation shows that the ordered pair $(3, -2)$ does not satisfy both equations simultaneously. Therefore, the correct answer is False.

Question 7

Solve the system of equations by the substitution method.
-$\left\{ \begin{array} { l } 
\frac { 1 } { 2 } x + 3 y = - \frac { 23 } { 4 } \
4 x = - y
\end{array} \right.$

Accepted Answer

The answer of Solve the system of equations by the...

Question 8

Answer true or false.
-If the two equations in a system of linear equations are added and the result is 7x = 0, the system has no
solution.

Accepted Answer

If the result of adding the two equations is 7x = 0, then it means that the two equations are parallel (i.e. they have the same slope but different y-intercepts). Since parallel lines never intersect, the system has no solution. If the result of adding the equations is 7x = 0, it means that both equations have the same slope and y-intercept. This can happen if the two equations are equivalent, meaning they represent the same line, and therefore have infinitely many solutions. If the two equations are parallel, they will never intersect and there will be no solution. However, in this case, we cannot determine whether the equations are parallel or equivalent, so we cannot conclude that there is no solution.

Question 9

Solve the system by graphing.
-$\left\{ \begin{array} { l } 
y = - 3 x \
3 x + y = 15
\end{array} \right.$

Accepted Answer

The answer of Solve the system by graphing.
-\(\left\{ \begin{array} {...

Question 10

Solve the system of equations by the addition method.
-$\left\{ \begin{array} { l } 
8 x - 12 y = 14 \
- 2 x + 3 y = 0
\end{array} \right.$

Accepted Answer

The answer of Solve the system of equations by the...

Question 11

Answer true or false.
-A system of two linear equations in two variables can have exactly two solutions.

Accepted Answer

A system of two linear equations in two variables can have either one solution, infinite solutions, or no solution. It cannot have exactly two solutions. A system of two linear equations in two variables can have either no solution, one solution, or infinitely many solutions, but not exactly two solutions.

Question 12

Solve the system of equations by the addition method.
-$\left\{ \begin{array} { l } 
x + y = 33 \
x - y = 13
\end{array} \right.$

Accepted Answer

Adding the equations gives $2x = 46$, so $x = 23$. Substituting $x = 23$ into the first equation gives $23 + y = 33$, so $y = 10$. Thus, the solution is (23, 10).

Question 13

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-Two or more linear equations are called a(n)_____________.

Accepted Answer

A system of linear equations refers to two or more linear equations that are to be solved together. A system of linear equations is a set of two or more linear equations that are to be solved simultaneously. This is the correct term to describe the situation given in the question. The other choices (dependent, solution, substitution, addition, consistent, independent, and inconsistent) are all important concepts related to systems of linear equations, but they do not accurately describe what a system of linear equations is.

Question 14

Is the ordered pair a solution of the linear system?
-$\left\{ \begin{array} { l } 
2 x - 4 y = 6 \
5 x + y = 1
\end{array} ; ( 1 , - 1 ) \right.$

Accepted Answer

The ordered pair (1, -1) does not satisfy both equations in the system. Substituting x = 1 and y = -1 into the first equation gives 2(1) - 4(-1) = 6, which is true. However, substituting into the second equation gives 5(1) + (-1) = 4, which is not equal to 1.

Question 15

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-A system of equations that has at least one solution is called a(n)_____________system.

Accepted Answer

A system of equations with at least one solution is called a consistent system. The other options are not accurate descriptions of a system with at least one solution. A system of equations that has at least one solution is a consistent system. Choices A and C refer to types of solutions within a system (independent and dependent, respectively), while choice B refers to a system with no solutions (inconsistent). Substitution and addition are methods for solving systems of equations.

Question 16

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-A system of equations that has no solution is called a(n)_____________system.

Accepted Answer

An inconsistent system of equations has no solution. Dependent, independent, and consistent systems all have at least one solution. Substitution and addition are methods for solving systems of linear equations, but they do not describe the nature of the solutions. A system of equations that has no solution is called inconsistent.

Question 17

Solve the system by graphing.
-$\left\{ \begin{array} { l } 
x - y = - 2 \
3 x - y = - 10
\end{array} \right.$

Accepted Answer

The answer of Solve the system by graphing.
-\(\left\{ \begin{array} {...

Question 18

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-A(n)_____________of a system of two equations in two variables is an ordered pair of numbers that is a solution of both equations in the system.

Accepted Answer

A solution of a system of linear equations is an ordered pair that satisfies both equations in the system. A solution of a system of equations is a pair of numbers that satisfy both equations in the system. The other options are related to types of systems and methods for solving them, but do not directly relate to the definition of a solution.

Question 19

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-In a system of linear equations in two variables, if the graphs of the equations are different, the equations are_____________ equations.

Accepted Answer

If the graphs of the equations are different, it means they do not intersect at any point, which means there is no solution. This makes the system inconsistent. However, if the equations are independent, they will have exactly one solution as long as the system is consistent. Therefore, the best choice is C. If the graphs of the equations are different, it means that they do not intersect, which indicates that there is no common solution. Therefore, the system is inconsistent. Since the system only has 2 variables, it can either be independent or dependent, and it cannot be solved using addition or substitution. However, we cannot determine whether the system is dependent or independent without additional information. Therefore, the best choice is C, which is independent, as it is the stronger condition of the two.

Question 20

Fill in the blank with one of the words or phrases listed below.
system of linear equations solution consistent independent
dependent inconsistent substitution addition
-Two algebraic methods for solving systems of equations are_____________ and_____________ .

Accepted Answer

The two algebraic methods for solving systems of equations are substitution and addition (also known as elimination). The terms "dependent" and "independent" are used to describe the relationship between equations in a system, and the terms "solution" and "inconsistent" describe the possible outcomes of solving a system. The two algebraic methods for solving systems of equations are addition and substitution. Dependent and independent refer to the nature of the solutions and whether a unique solution exists. Solution is not a method for solving, but rather the result of finding the values for the variables. Inconsistent refers to a system with no solution, which is not a method for solving.

Quiz 11: Systems of Equations

Answer true or false. -If the two equations in a system of linear equations are added and the result is 8 = 0, the system has no solution.

Answer true or false. -Although (7, 3)is not a solution of x + 3y = 2, it can still be a solution of the system $\left\{ \begin{array} { l } x + 3 y = 2 \\x + y = 4\end{array} \right.$

Solve the system of equations by the substitution method. - $\left\{ \begin{array} { l } 4 x - 3 y = - 15 \\y = x + 4\end{array} \right.$

Solve the system by the substitution or the addition method. - $\left\{ \begin{array} { l } 3 x + y = 4 \\4 x + 3 y = - 3\end{array} \right.$

Is the ordered pair a solution of the linear system? - $\left\{ \begin{array} { l } 5 x - 3 y = 21 \\4 x + 5 y = - 3\end{array} ; \quad ( 3 , - 2 ) \right.$

Solve the system of equations by the substitution method. - $\left\{ \begin{array} { l } \frac { 1 } { 2 } x + 3 y = - \frac { 23 } { 4 } \\4 x = - y\end{array} \right.$

Answer true or false. -If the two equations in a system of linear equations are added and the result is 7x = 0, the system has no solution.

Solve the system by graphing. - $\left\{ \begin{array} { l } y = - 3 x \\3 x + y = 15\end{array} \right.$ $Solve the system by graphing. - \left\{ \begin{array} { l } y = - 3 x \\ 3 x + y = 15 \end{array} \right. A) (3, 15) B) (0, 0) C) Infinite number of solutions D) No solution$

Solve the system of equations by the addition method. - $\left\{ \begin{array} { l } 8 x - 12 y = 14 \\- 2 x + 3 y = 0\end{array} \right.$

Answer true or false. -A system of two linear equations in two variables can have exactly two solutions.

Solve the system of equations by the addition method. - $\left\{ \begin{array} { l } x + y = 33 \\x - y = 13\end{array} \right.$

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -Two or more linear equations are called a(n) _____________.

Is the ordered pair a solution of the linear system? - $\left\{ \begin{array} { l } 2 x - 4 y = 6 \\5 x + y = 1\end{array} ; ( 1 , - 1 ) \right.$

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -A system of equations that has at least one solution is called a(n) _____________system.

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -A system of equations that has no solution is called a(n) _____________system.

Solve the system by graphing. - $\left\{ \begin{array} { l } x - y = - 2 \\3 x - y = - 10\end{array} \right.$ $Solve the system by graphing. - \left\{ \begin{array} { l } x - y = - 2 \\ 3 x - y = - 10 \end{array} \right. A) (0, -2) B) (-5, -1) C) (-2, 4) D) (-4, -2)$

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -Two algebraic methods for solving systems of equations are_ and_ .

Quiz 11: Systems of Equations

Answer true or false. -If the two equations in a system of linear equations are added and the result is 8 = 0, the system has no solution.

Answer true or false. -Although (7, 3)is not a solution of x + 3y = 2, it can still be a solution of the system {x+3y=2x+y=4\left\{ \begin{array} { l } x + 3 y = 2 \\x + y = 4\end{array} \right.{x+3y=2x+y=4​

Solve the system of equations by the substitution method. - {4x−3y=−15y=x+4\left\{ \begin{array} { l } 4 x - 3 y = - 15 \\y = x + 4\end{array} \right.{4x−3y=−15y=x+4​

Solve the system by the substitution or the addition method. - {3x+y=44x+3y=−3\left\{ \begin{array} { l } 3 x + y = 4 \\4 x + 3 y = - 3\end{array} \right.{3x+y=44x+3y=−3​

Is the ordered pair a solution of the linear system? - {5x−3y=214x+5y=−3;(3,−2)\left\{ \begin{array} { l } 5 x - 3 y = 21 \\4 x + 5 y = - 3\end{array} ; \quad ( 3 , - 2 ) \right.{5x−3y=214x+5y=−3​;(3,−2)

Solve the system of equations by the substitution method. - {12x+3y=−2344x=−y\left\{ \begin{array} { l } \frac { 1 } { 2 } x + 3 y = - \frac { 23 } { 4 } \\4 x = - y\end{array} \right.{21​x+3y=−423​4x=−y​

Answer true or false. -If the two equations in a system of linear equations are added and the result is 7x = 0, the system has no solution.

Solve the system by graphing. - {y=−3x3x+y=15\left\{ \begin{array} { l } y = - 3 x \\3 x + y = 15\end{array} \right.{y=−3x3x+y=15​

Solve the system of equations by the addition method. - {8x−12y=14−2x+3y=0\left\{ \begin{array} { l } 8 x - 12 y = 14 \\- 2 x + 3 y = 0\end{array} \right.{8x−12y=14−2x+3y=0​

Answer true or false. -A system of two linear equations in two variables can have exactly two solutions.

Solve the system of equations by the addition method. - {x+y=33x−y=13\left\{ \begin{array} { l } x + y = 33 \\x - y = 13\end{array} \right.{x+y=33x−y=13​

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -Two or more linear equations are called a(n) _____________.

Is the ordered pair a solution of the linear system? - {2x−4y=65x+y=1;(1,−1)\left\{ \begin{array} { l } 2 x - 4 y = 6 \\5 x + y = 1\end{array} ; ( 1 , - 1 ) \right.{2x−4y=65x+y=1​;(1,−1)

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -A system of equations that has at least one solution is called a(n) _____________system.

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -A system of equations that has no solution is called a(n) _____________system.

Solve the system by graphing. - {x−y=−23x−y=−10\left\{ \begin{array} { l } x - y = - 2 \\3 x - y = - 10\end{array} \right.{x−y=−23x−y=−10​

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -Two algebraic methods for solving systems of equations are_____________ and_____________ .

Answer true or false. -Although (7, 3)is not a solution of x + 3y = 2, it can still be a solution of the system $\left\{ \begin{array} { l } x + 3 y = 2 \\x + y = 4\end{array} \right.$

Solve the system of equations by the substitution method. - $\left\{ \begin{array} { l } 4 x - 3 y = - 15 \\y = x + 4\end{array} \right.$

Solve the system by the substitution or the addition method. - $\left\{ \begin{array} { l } 3 x + y = 4 \\4 x + 3 y = - 3\end{array} \right.$

Is the ordered pair a solution of the linear system? - $\left\{ \begin{array} { l } 5 x - 3 y = 21 \\4 x + 5 y = - 3\end{array} ; \quad ( 3 , - 2 ) \right.$

Solve the system of equations by the substitution method. - $\left\{ \begin{array} { l } \frac { 1 } { 2 } x + 3 y = - \frac { 23 } { 4 } \\4 x = - y\end{array} \right.$

Solve the system by graphing. - $\left\{ \begin{array} { l } y = - 3 x \\3 x + y = 15\end{array} \right.$ $Solve the system by graphing. - \left\{ \begin{array} { l } y = - 3 x \\ 3 x + y = 15 \end{array} \right. A) (3, 15) B) (0, 0) C) Infinite number of solutions D) No solution$

Solve the system of equations by the addition method. - $\left\{ \begin{array} { l } 8 x - 12 y = 14 \\- 2 x + 3 y = 0\end{array} \right.$

Solve the system of equations by the addition method. - $\left\{ \begin{array} { l } x + y = 33 \\x - y = 13\end{array} \right.$

Is the ordered pair a solution of the linear system? - $\left\{ \begin{array} { l } 2 x - 4 y = 6 \\5 x + y = 1\end{array} ; ( 1 , - 1 ) \right.$

Solve the system by graphing. - $\left\{ \begin{array} { l } x - y = - 2 \\3 x - y = - 10\end{array} \right.$ $Solve the system by graphing. - \left\{ \begin{array} { l } x - y = - 2 \\ 3 x - y = - 10 \end{array} \right. A) (0, -2) B) (-5, -1) C) (-2, 4) D) (-4, -2)$

Fill in the blank with one of the words or phrases listed below. system of linear equations solution consistent independent dependent inconsistent substitution addition -Two algebraic methods for solving systems of equations are_ and_ .