Question 1

Use the addition method to solve the system. $$\left\{ \begin{array} { c } 
a + b = 5 \
a - b = - 1
\end{array} \right.$$

Accepted Answer

Using the addition method, add the left-hand sides of the two equations to get $a+b+a-b=2a=4 \implies a=2$. Substitute this value back into either equation to solve for $b$: $2+b=5 \implies b=3$. Therefore, the solution to the system is $(2,3)$, which corresponds to choice C.

Question 2

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { c } 
3 ( x - 4 ) = 8 y \
4 ( 2 y + 3 ) = 3 x
\end{array} \right.$$

Accepted Answer

Multiply the first equation by $3$ and the second equation by $2$, then we get: $$\left\{ \begin{array} { l }9x-24=24y \ 6x+8y+9x=0\end{array} \right.$$
Adding these two equations gives $24x=24y$, or $y=x$. Substitute into either equation gives $x=3$ and $y=3$, thus the equations are dependent and have infinite solutions, so the correct answer is $\textbf{(D)}$.

Question 3

Use two equations in two variables to solve the following problem. A rancher raises five times as many cows as horses. If he has 144 animals (cows and horses only), how many cows does he have?

Accepted Answer

Let the number of horses be $x$ and the number of cows be $5x$. The total number of animals is $x + 5x = 144$. Solving this equation gives $6x = 144$, so $x = 24$. Therefore, the number of cows is $5 \times 24 = 120$.

Question 4

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $\left\{ \begin{array} { c } 
0.04 x - 0.02 y = 0.08 \
2 x - 4 = y
\end{array} \right.$

Accepted Answer

The answer of Solve the system by any method, if...

Question 5

Use the method of elimination to find the y -coordinate of the solution of the following system of equations. $$\left\{ \begin{array} { l } 
4 x - 5 y = - 32 \
2 x + 5 y = 14
\end{array} \right.$$

Accepted Answer

The answer of Use the method of elimination to find...

Question 6

Solve the system by elimination if possible. $$\left\{ \begin{array} { c } 
2 x + 3 y = 29 \
3 x - 2 y = - 15
\end{array} \right.$$

Accepted Answer

To solve the system by elimination, we can multiply the equations by appropriate values to make the coefficients of either $x$ or $y$ cancel out when we add or subtract the equations. Multiplying the first equation by 2 and the second equation by 3 gives us:$$4x + 6y = 58$$$$9x - 6y = -45$$Adding these equations together to eliminate $y$, we get:$$13x = 13$$Solving for $x$, we find $x = 1$. Substituting $x = 1$ into the first original equation:$$2(1) + 3y = 29$$$$2 + 3y = 29$$$$3y = 27$$$$y = 9$$Therefore, the solution to the system is $x = 1$, $y = 9$, which corresponds to choice A.

Question 7

A tree casts a shadow of 26 feet at the same time as a 6-foot man casts a shadow of 4 feet. Find the height of the tree.

Accepted Answer

We can use proportions to solve the problem. Let x be the height of the tree. Then, we have:
(x/26) = (6/4)
Cross-multiplying, we get:
4x = 156
x = 39
Therefore, the height of the tree is 39 feet.

Question 8

In solving a system of equations by the elimination method, the result 1 = 1 is obtained. Which of the following statements is true?

Accepted Answer

The answer of In solving a system of equations by...

Question 9

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $$\left\{ \begin{array} { l } 
x = \frac { 3 } { 2 } y + 9 \
2 x - 3 y = 2
\end{array} \right.$$

Accepted Answer

The answer of Solve the system by any method, if...

Question 10

Use two equations in two variables to solve the following problem. Maria and Susan pool their resources to buy several lottery tickets. They win $700,000! They agree that Susan should get $50,000 more than Maria, because she gave most of the money. How much will Maria get?

Accepted Answer

The answer of Use two equations in two variables to...

Question 11

Solve the system by any method, if possible. $$\left\{ \begin{array} { l } 
x = \frac { 3 } { 2 } y + 3 \
2 x - 3 y = 9
\end{array} \right.$$

Accepted Answer

The answer of Solve the system by any method, if...

Question 12

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { l } 
8 x - y = - 25 \
9 x + 6 y = - 21
\end{array} \right)$$

Accepted Answer

Substituting x = -3 into the first equation gives y = 1, which satisfies both equations.

Question 13

A box contains $7.85 in nickels, dimes, and quarters. There are 48 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there?

Accepted Answer

The answer of A box contains $7.85 in nickels, dimes,...

Question 14

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 
2 ( x + 2 ) + 9 ( y - 4 ) = 1 \
2 ( x - 1 ) = - 9 ( y + 2 )
\end{array} \right.$$

Accepted Answer

First, simplify and expand both equations: 1) $2(x + 2) + 9(y - 4) = 1$ becomes $2x + 4 + 9y - 36 = 1$, which simplifies to $2x + 9y - 32 = 0$.2) $2(x - 1) = -9(y + 2)$ becomes $2x - 2 = -9y - 18$, which simplifies to $2x + 9y + 16 = 0$.Multiplying the first equation by 1 and the second equation by 1 does not eliminate any variable when added together, indicating the system might be inconsistent or dependent. Upon closer inspection, there is no scalar multiple that makes the two equations identical (which would indicate dependence) or results in a contradiction (like $0 = 1$, which would indicate inconsistency). However, rearranging the terms and comparing the equations:1) $2x + 9y = 32$2) $2x + 9y = -16$These two equations cannot be true at the same time since they assign two different constants to the same linear combination of $x$ and $y$, indicating the system is inconsistent.

Question 15

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } 
3 x - 8 y = 3 \
6 x - 16 y = - 2
\end{array} \right)$$

Accepted Answer

The answer of Solve the system by using either the...

Question 16

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } 
\frac { 1 } { 2 } x - \frac { 1 } { 6 } y = - 12 \
\frac { 1 } { 2 } x + \frac { 1 } { 4 } y = 8
\end{array} \right)$$

Accepted Answer

The answer of Solve the system by using either the...

Question 17

Use the dimensions in the illustration to find w , the width of the river. The two triangles in the illustration are similar.

Accepted Answer

The answer of Use the dimensions in the illustration to...

Question 18

If the addition method is used to solve the following system, by what would we multiply the first equation to eliminate m ? $$\left\{ \begin{aligned}
6 m + 3 y & = 9 \
48 m - 10 & = 18
\end{aligned} \right.$$

Accepted Answer

To eliminate $m$ using the addition method, we need to make the coefficients of $m$ in both equations opposites of each other. The first equation has a coefficient of $6$ for $m$, and the second equation has a coefficient of $48$ for $m$. To make these coefficients opposites, we can multiply the first equation by $-8$. This gives $-8 \times 6m = -48m$, which will cancel out the $48m$ in the second equation when added together.

Question 19

Use two equations in two variables to solve the following problem. At a theater, the giant rectangular movie screen has a width $24$ feet less than its length. If its perimeter is $308$ feet, find the area of the screen.

Accepted Answer

The answer of Use two equations in two variables to...

Question 20

Use two equations in two variables to solve the following problem. In a year the number of people dying from cancer was 6 times the number who died from accidents. If the number of deaths from these two causes totaled 280,000, how many people died from each cause?

Accepted Answer

The answer of Use two equations in two variables to...

Quiz 4: Solving Systems of Equations and Inequalities

Use the addition method to solve the system. $\left\{ \begin{array} { c } a + b = 5 \\a - b = - 1\end{array} \right.$

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $\left\{ \begin{array} { c } 3 ( x - 4 ) = 8 y \\4 ( 2 y + 3 ) = 3 x\end{array} \right.$

Use two equations in two variables to solve the following problem. A rancher raises five times as many cows as horses. If he has 144 animals (cows and horses only) , how many cows does he have?

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $\left\{ \begin{array} { c } 0.04 x - 0.02 y = 0.08 \\2 x - 4 = y\end{array} \right.$

Use the method of elimination to find the y -coordinate of the solution of the following system of equations. $\left\{ \begin{array} { l } 4 x - 5 y = - 32 \\2 x + 5 y = 14\end{array} \right.$

Solve the system by elimination if possible. $\left\{ \begin{array} { c } 2 x + 3 y = 29 \\3 x - 2 y = - 15\end{array} \right.$

A tree casts a shadow of 26 feet at the same time as a 6-foot man casts a shadow of 4 feet. Find the height of the tree.

In solving a system of equations by the elimination method, the result 1 = 1 is obtained. Which of the following statements is true?

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 9 \\2 x - 3 y = 2\end{array} \right.$

Use two equations in two variables to solve the following problem. Maria and Susan pool their resources to buy several lottery tickets. They win $700,000! They agree that Susan should get $50,000 more than Maria, because she gave most of the money. How much will Maria get?

Solve the system by any method, if possible. $\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 3 \\2 x - 3 y = 9\end{array} \right.$

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { l } 8 x - y = - 25 \\9 x + 6 y = - 21\end{array} \right)$

A box contains $7.85 in nickels, dimes, and quarters. There are 48 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there?

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $\left\{ \begin{array} { l } 2 ( x + 2 ) + 9 ( y - 4 ) = 1 \\2 ( x - 1 ) = - 9 ( y + 2 ) \end{array} \right.$

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c } 3 x - 8 y = 3 \\6 x - 16 y = - 2\end{array} \right)$

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c } \frac { 1 } { 2 } x - \frac { 1 } { 6 } y = - 12 \\\frac { 1 } { 2 } x + \frac { 1 } { 4 } y = 8\end{array} \right)$

If the addition method is used to solve the following system, by what would we multiply the first equation to eliminate m ? $\left\{ \begin{aligned}6 m + 3 y & = 9 \\48 m - 10 & = 18\end{aligned} \right.$

Use two equations in two variables to solve the following problem. At a theater, the giant rectangular movie screen has a width $24$ feet less than its length. If its perimeter is $308$ feet, find the area of the screen.

Use two equations in two variables to solve the following problem. In a year the number of people dying from cancer was 6 times the number who died from accidents. If the number of deaths from these two causes totaled 280,000, how many people died from each cause?

Quiz 4: Solving Systems of Equations and Inequalities

Use the addition method to solve the system. {a+b=5a−b=−1\left\{ \begin{array} { c } a + b = 5 \\a - b = - 1\end{array} \right.{a+b=5a−b=−1​

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. {3(x−4) =8y4(2y+3) =3x\left\{ \begin{array} { c } 3 ( x - 4 ) = 8 y \\4 ( 2 y + 3 ) = 3 x\end{array} \right.{3(x−4) =8y4(2y+3) =3x​

Use two equations in two variables to solve the following problem. A rancher raises five times as many cows as horses. If he has 144 animals (cows and horses only) , how many cows does he have?

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. {0.04x−0.02y=0.082x−4=y\left\{ \begin{array} { c } 0.04 x - 0.02 y = 0.08 \\2 x - 4 = y\end{array} \right.{0.04x−0.02y=0.082x−4=y​

Use the method of elimination to find the y -coordinate of the solution of the following system of equations. {4x−5y=−322x+5y=14\left\{ \begin{array} { l } 4 x - 5 y = - 32 \\2 x + 5 y = 14\end{array} \right.{4x−5y=−322x+5y=14​

Solve the system by elimination if possible. {2x+3y=293x−2y=−15\left\{ \begin{array} { c } 2 x + 3 y = 29 \\3 x - 2 y = - 15\end{array} \right.{2x+3y=293x−2y=−15​

A tree casts a shadow of 26 feet at the same time as a 6-foot man casts a shadow of 4 feet. Find the height of the tree.

In solving a system of equations by the elimination method, the result 1 = 1 is obtained. Which of the following statements is true?

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. {x=32y+92x−3y=2\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 9 \\2 x - 3 y = 2\end{array} \right.{x=23​y+92x−3y=2​

Use two equations in two variables to solve the following problem. Maria and Susan pool their resources to buy several lottery tickets. They win $700,000! They agree that Susan should get $50,000 more than Maria, because she gave most of the money. How much will Maria get?

Solve the system by any method, if possible. {x=32y+32x−3y=9\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 3 \\2 x - 3 y = 9\end{array} \right.{x=23​y+32x−3y=9​

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. (8x−y=−259x+6y=−21) \left( \begin{array} { l } 8 x - y = - 25 \\9 x + 6 y = - 21\end{array} \right) (8x−y=−259x+6y=−21​)

A box contains $7.85 in nickels, dimes, and quarters. There are 48 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there?

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. (3x−8y=36x−16y=−2) \left( \begin{array} { c } 3 x - 8 y = 3 \\6 x - 16 y = - 2\end{array} \right) (3x−8y=36x−16y=−2​)

Use the dimensions in the illustration to find w , the width of the river. The two triangles in the illustration are similar.

If the addition method is used to solve the following system, by what would we multiply the first equation to eliminate m ? {6m+3y=948m−10=18\left\{ \begin{aligned}6 m + 3 y & = 9 \\48 m - 10 & = 18\end{aligned} \right.{6m+3y48m−10​=9=18​

Use two equations in two variables to solve the following problem. At a theater, the giant rectangular movie screen has a width 242424 feet less than its length. If its perimeter is 308308308 feet, find the area of the screen.

Use two equations in two variables to solve the following problem. In a year the number of people dying from cancer was 6 times the number who died from accidents. If the number of deaths from these two causes totaled 280,000, how many people died from each cause?

Use the addition method to solve the system. $\left\{ \begin{array} { c } a + b = 5 \\a - b = - 1\end{array} \right.$

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $\left\{ \begin{array} { c } 3 ( x - 4 ) = 8 y \\4 ( 2 y + 3 ) = 3 x\end{array} \right.$

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $\left\{ \begin{array} { c } 0.04 x - 0.02 y = 0.08 \\2 x - 4 = y\end{array} \right.$

Use the method of elimination to find the y -coordinate of the solution of the following system of equations. $\left\{ \begin{array} { l } 4 x - 5 y = - 32 \\2 x + 5 y = 14\end{array} \right.$

Solve the system by elimination if possible. $\left\{ \begin{array} { c } 2 x + 3 y = 29 \\3 x - 2 y = - 15\end{array} \right.$

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate. $\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 9 \\2 x - 3 y = 2\end{array} \right.$

Solve the system by any method, if possible. $\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 3 \\2 x - 3 y = 9\end{array} \right.$

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { l } 8 x - y = - 25 \\9 x + 6 y = - 21\end{array} \right)$

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c } 3 x - 8 y = 3 \\6 x - 16 y = - 2\end{array} \right)$

If the addition method is used to solve the following system, by what would we multiply the first equation to eliminate m ? $\left\{ \begin{aligned}6 m + 3 y & = 9 \\48 m - 10 & = 18\end{aligned} \right.$

Use two equations in two variables to solve the following problem. At a theater, the giant rectangular movie screen has a width $24$ feet less than its length. If its perimeter is $308$ feet, find the area of the screen.