Question 1

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } 
- 8 x + 7 y = - 11 \
16 x + 14 y = 106
\end{array} \right.$$ ​
A)(​4,3)
B)​(16,14)
C)​​(- 4,- 3)
D)​(-11,106)
E)​(- 16,- 14)

Accepted Answer

Substitute x = 4 and y = 3 into both equations. Both equations are satisfied, confirming (4,3) is the solution.

Question 2

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 48 meters and the length is 2 meters more than the width.
A)13 m × 11 m
B)16 m × 14 m
C)16 m × 8 m
D)17 m × 15 m
E)12 m × 12 m

Accepted Answer

The perimeter of a rectangle is given by P = 2(l + w), where P is the perimeter, l is the length, and w is the width. We are given that the perimeter is 48 meters, so we can write:48 = 2(l + w)24 = l + wWe are also given that the length is 2 meters more than the width, so we can write:l = w + 2Substituting this into the first equation, we get:24 = (w + 2) + w24 = 2w + 222 = 2ww = 11Substituting this back into the equation l = w + 2, we get:l = 11 + 2l = 13Therefore, the dimensions of the rectangle are 13 meters by 11 meters.

Question 3

Find the sales necessary to break even (R - C = 0)for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. )​ $$C = 6.7 \sqrt { x } + 3000 , R = 8.1 x$$ ​
A)No real solution
B)387 units
C)355 units
D)377 units
E)355 units or 387 units

Accepted Answer

To find the break-even point, set $R = C$ and solve for $x$. Thus, $8.1x = 6.7\sqrt{x} + 3000$. Rearranging and solving this equation for $x$ gives us the break-even point in units. After solving, the closest whole unit for the break-even point is 387 units.

Question 4

Determine which ordered pair is a solution of the system.​ $$\left\{ \begin{array} { l } 
- 8 x + 7 y = - 10 \
- 9 x + 4 y = 12
\end{array} \right.$$ ​
A)(3,-1)
B)(4,-6)
C)(-4,-6)
D)(-6,-4)
E)(-1,-3)

Accepted Answer

The answer of Determine which ordered pair is a solution...

Question 5

Solve the system of equations by graphing.​ $$\left\{ \begin{array} { l } 
4 x - 9 y = - 60 \
2 x + 9 y = 78
\end{array} \right.$$ ​
A)(11,-4)
B)(-3,-8)
C)(3,8)
D)(-5,12)
E)(8,3)

Accepted Answer

By graphing the system of equations, we find that the two lines intersect at the point (3,8). This is the only point that satisfies both equations, and therefore is the solution to the system. The other answer choices do not lie on both lines and do not satisfy both equations.

Question 6

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}
x - y & = - 22 \
x ^ { 2 } - y & = - 2
\end{aligned} \right.$$ ​
A)(5,-13), (-4,-22)
B)No real solution
C)(5,-17), (-4,-26)
D)(5,27), (-4,18)
E)(5,-13)

Accepted Answer

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

Question 7

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } 
- 6 x + 7 y = 50 \
12 x + 14 y = 124
\end{array} \right.$$ ​
A)(9,-6)
B)(1,8)
C)(-6,9)
D)(-7,10)
E)No solutions

Accepted Answer

Substitute $ x = 1 $ and $ y = 8 $ into both equations:  First equation: $-6(1) + 7(8) = -6 + 56 = 50$  Second equation: $12(1) + 14(8) = 12 + 112 = 124$  Both equations are satisfied, so (1,8) is the solution.

Question 8

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } 
y = - 4 x \
y = x ^ { 3 } + x ^ { 2 } - x
\end{array} \right.$$ ​
A)(3,4)
B)(0,3)
C)(3,4), (-3,-4)
D)No real solution
E)(0,0)

Accepted Answer

Substitute $ y = -4x $ into $ y = x^3 + x^2 - x $ to get $-4x = x^3 + x^2 - x$. Simplifying gives $x^3 + x^2 + 3x = 0$, which factors to $x(x^2 + x + 3) = 0$. The real solution is $x = 0$, giving $y = -4(0) = 0$, so the solution is (0,0).

Question 9

​Solve the system by the method of substitution. ​​ $$\left\{ \begin{array} { l } 
\frac { 1 } { 10 } x + \frac { 3 } { 10 } y = 1 \\
- \frac { 1 } { 5 } x + \frac { 2 } { 5 } y = 1
\end{array} \right.$$ ​
A)​(-1,-3)
B)​(-3,1)
C)​(1,-3)
D)​(3,1)
E)​(1,3)

Accepted Answer

The answer of ​Solve the system by the method of...

Question 10

Solve the system.​ $$\left\{ \begin{array} { l } 
12 x - 28 y = 12 \
- 6 x + 14 y = - 6
\end{array} \right.$$ ​
A)x = 3,y = 7
B)x = 4,y = 3
C) $$\left( x , \frac { 3 x } { 7 } - \frac { 3 } { 7 } \right)$$
D)x = 2,y = 3
E)No solution

Accepted Answer

The second equation is exactly half of the first, indicating the system has infinitely many solutions. The correct answer expresses y in terms of x, fitting the criteria for an infinite solution set.

Question 11

Determine which ordered pair is a solution of the system.​ $$\left\{ \begin{array} { l } 
- x + 9 y ^ { 2 } = 12 \
- 4 x + 9 y = 3
\end{array} \right.$$ ​
A)(-3,1)
B)(-8,7)
C)(-1,3)
D)(-3,-1)
E)(-8,8)

Accepted Answer

The answer of Determine which ordered pair is a solution...

Question 12

Solve the system of equations by graphing. ​​ $$\left\{ \begin{aligned}
x + y & = 0 \
6 x - 7 y & = 13
\end{aligned} \right.$$ ​
A)(-2,1)
B)(-1,1)
C)(0,2)
D)(1,-1)
E)(3,5)

Accepted Answer

By graphing the two equations, we can see that they intersect at the point (1,-1). This is the only point that satisfies both equations, so the answer is (D) (1,-1).

Question 13

Use a graphing utility to solve the system of equations.Find the solution accurate to two decimal places.​ $$\left\{ \begin{array} { l } 
y = 2 e ^ { - x } \
y = \ln ( x - 2 ) + 1
\end{array} \right.$$ ​
A)​(0.13,1.76)
B)​(- 1.40,0.49)and (- 0.25,1.56)
C)no real solution
D)​(4.77,0.07)
E)​(2.44,0.17)

Accepted Answer

The solution to the system of equations is the point where the graphs of the equations intersect. Using a graphing utility, the intersection point is approximately (2.44, 0.17).

Question 14

The sales of various types of lawn and garden tools vary according to the season.At a certain home improvement store,the monthly sales H of garden hoes (hoes sold per month)decline from July to October whereas the monthly sales of lawn rakes R (rakes sold per month)increase during this same interval.The sales of these two items during the calendar months July-October are modeled by the equations:​ $$\begin{array} { l } 
H ( t ) = 64 - 6 t \
R ( t ) = 17 t - 143
\end{array}$$ ​ where t is the month (t = 7 corresponds to July).In which month does the number of rakes sold equal the number of hoes sold?
​
A)September
B)July
C)November
D)October
E)August

Accepted Answer

To find the month when the number of rakes sold equals the number of hoes sold, set $H(t) = R(t)$ and solve for $t$:$$64 - 6t = 17t - 143$$$$207 = 23t$$$$t = 9$$Since $t = 9$ corresponds to September, the correct answer is September.

Question 15

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } 
y = 7 x - 21 \
9 x - y = 29
\end{array} \right.$$ ​
A)(11,-2)
B)(7,4)
C)(-3,12)
D)(4,7)
E)No solutions

Accepted Answer

Substituting $y = 7x - 21$ into the second equation gives $9x - (7x - 21) = 29$. Simplifying, we get $2x + 21 = 29$, leading to $2x = 8$, and thus $x = 4$. Substituting $x = 4$ into $y = 7x - 21$ gives $y = 7(4) - 21 = 28 - 21 = 7$. Therefore, the solution is $x = 4, y = 7$.

Question 16

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } 
4 x + 3 y = 23 \
8 x + 6 y = 24
\end{array} \right.$$ ​
A)(6,5)
B)(4,7)
C)(-5,-1)
D)(2,5)
E)No solution;inconsistent system

Accepted Answer

The given system of equations is actually a case of parallel lines, which can be seen by simplifying the second equation. Dividing the second equation by 2 gives $4x + 3y = 12$, which is not consistent with the first equation $4x + 3y = 23$. This inconsistency means that there is no point (x, y) that satisfies both equations simultaneously, indicating that the system of equations has no solution.

Question 17

​Solve the system of equations by graphing.​ $$\left\{ \begin{array} { l } 
2 x + 3 y = 28 \
6 x + 5 y = 60
\end{array} \right.$$ ​ ​
A)(-6,5)
B)​(3,- 5)
C)​(5,6)
D)​(2,6)
E)​(5,3)

Accepted Answer

To solve the system of equations by graphing, we first rearrange each equation into slope-intercept form (y = mx + b). For the first equation, $2x + 3y = 28$, we get $y = -\frac{2}{3}x + \frac{28}{3}$. For the second equation, $6x + 5y = 60$, we get $y = -\frac{6}{5}x + 12$. Graphing these two lines, we look for the point where they intersect. The correct intersection point that satisfies both equations is (5,6), which corresponds to choice C.

Question 18

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } 
y = x ^ { 3 } - 6 x ^ { 2 } + 4 \
y = x ^ { 2 } - 10 x + 4
\end{array} \right.$$ ​
A)(2,-12), (1,-5)
B)(5,-21), (2,-12), (0,4)
C)(5,-21), (-1,15)
D)No real solution
E)(5,-21), (-2,28), (1,-5)

Accepted Answer

1) Solve for y in the first equation: 
2x + y = -2 
y = -2x - 2 
2) Substitute y=-2x-2 into the second equation: 
x^2 + (-2x-2)^2 = 25 
Simplify and solve for x: 
5x^2 + 8x - 9 = 0 
(5x - 9)(x + 1) = 0 
x = 9/5 or x = -1 
3) Substitute each value of x back into y=-2x-2 to find the corresponding y-value: 
When x = 9/5, y = -21 
When x = -1, y = 4 
So the solution set is {(9/5,-21), (-1,4)}. Option B lists these solutions, so it is the best choice.

Quiz 41: Linear and Nonlinear Systems of Equations

Solve the system by substitution,if possible. $\left\{ \begin{array} { l } - 8 x + 7 y = - 11 \\16 x + 14 y = 106\end{array} \right.$

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 48 meters and the length is 2 meters more than the width.

Find the sales necessary to break even (R - C = 0) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. ) $C = 6.7 \sqrt { x } + 3000 , R = 8.1 x$

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } - 8 x + 7 y = - 10 \\- 9 x + 4 y = 12\end{array} \right.$

Solve the system of equations by graphing. $\left\{ \begin{array} { l } 4 x - 9 y = - 60 \\2 x + 9 y = 78\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{aligned}x - y & = - 22 \\x ^ { 2 } - y & = - 2\end{aligned} \right.$

Solve the system by substitution,if possible. $\left\{ \begin{array} { l } - 6 x + 7 y = 50 \\12 x + 14 y = 124\end{array} \right.$

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

Solve the system by the method of substitution. $\left\{ \begin{array} { l } \frac { 1 } { 10 } x + \frac { 3 } { 10 } y = 1 \\\\- \frac { 1 } { 5 } x + \frac { 2 } { 5 } y = 1\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } 12 x - 28 y = 12 \\- 6 x + 14 y = - 6\end{array} \right.$

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } - x + 9 y ^ { 2 } = 12 \\- 4 x + 9 y = 3\end{array} \right.$

Solve the system of equations by graphing. $\left\{ \begin{aligned}x + y & = 0 \\6 x - 7 y & = 13\end{aligned} \right.$

Use a graphing utility to solve the system of equations.Find the solution accurate to two decimal places. $\left\{ \begin{array} { l } y = 2 e ^ { - x } \\y = \ln ( x - 2 ) + 1\end{array} \right.$

Solve the system by substitution,if possible. $\left\{ \begin{array} { l } y = 7 x - 21 \\9 x - y = 29\end{array} \right.$

Solve the system by substitution,if possible. $\left\{ \begin{array} { l } 4 x + 3 y = 23 \\8 x + 6 y = 24\end{array} \right.$

Solve the system of equations by graphing. $\left\{ \begin{array} { l } 2 x + 3 y = 28 \\6 x + 5 y = 60\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{array} { l } y = x ^ { 3 } - 6 x ^ { 2 } + 4 \\y = x ^ { 2 } - 10 x + 4\end{array} \right.$

Quiz 41: Linear and Nonlinear Systems of Equations

Solve the system by substitution,if possible.​ {−8x+7y=−1116x+14y=106\left\{ \begin{array} { l } - 8 x + 7 y = - 11 \\16 x + 14 y = 106\end{array} \right.{−8x+7y=−1116x+14y=106​ ​

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 48 meters and the length is 2 meters more than the width.

Find the sales necessary to break even (R - C = 0) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. ) ​ C=6.7x+3000,R=8.1xC = 6.7 \sqrt { x } + 3000 , R = 8.1 xC=6.7x​+3000,R=8.1x ​

Determine which ordered pair is a solution of the system.​ {−8x+7y=−10−9x+4y=12\left\{ \begin{array} { l } - 8 x + 7 y = - 10 \\- 9 x + 4 y = 12\end{array} \right.{−8x+7y=−10−9x+4y=12​ ​

Solve the system of equations by graphing.​ {4x−9y=−602x+9y=78\left\{ \begin{array} { l } 4 x - 9 y = - 60 \\2 x + 9 y = 78\end{array} \right.{4x−9y=−602x+9y=78​ ​

Solve the system by the method of substitution.​ {x−y=−22x2−y=−2\left\{ \begin{aligned}x - y & = - 22 \\x ^ { 2 } - y & = - 2\end{aligned} \right.{x−yx2−y​=−22=−2​ ​

Solve the system by substitution,if possible.​ {−6x+7y=5012x+14y=124\left\{ \begin{array} { l } - 6 x + 7 y = 50 \\12 x + 14 y = 124\end{array} \right.{−6x+7y=5012x+14y=124​ ​

Solve the system by the method of substitution.​ {y=−4xy=x3+x2−x\left\{ \begin{array} { l } y = - 4 x \\y = x ^ { 3 } + x ^ { 2 } - x\end{array} \right.{y=−4xy=x3+x2−x​ ​

​Solve the system by the method of substitution. ​​ {110x+310y=1−15x+25y=1\left\{ \begin{array} { l } \frac { 1 } { 10 } x + \frac { 3 } { 10 } y = 1 \\\\- \frac { 1 } { 5 } x + \frac { 2 } { 5 } y = 1\end{array} \right.⎩⎨⎧​101​x+103​y=1−51​x+52​y=1​ ​

Solve the system.​ {12x−28y=12−6x+14y=−6\left\{ \begin{array} { l } 12 x - 28 y = 12 \\- 6 x + 14 y = - 6\end{array} \right.{12x−28y=12−6x+14y=−6​ ​

Determine which ordered pair is a solution of the system.​ {−x+9y2=12−4x+9y=3\left\{ \begin{array} { l } - x + 9 y ^ { 2 } = 12 \\- 4 x + 9 y = 3\end{array} \right.{−x+9y2=12−4x+9y=3​ ​

Solve the system of equations by graphing. ​​ {x+y=06x−7y=13\left\{ \begin{aligned}x + y & = 0 \\6 x - 7 y & = 13\end{aligned} \right.{x+y6x−7y​=0=13​ ​

Use a graphing utility to solve the system of equations.Find the solution accurate to two decimal places.​ {y=2e−xy=ln⁡(x−2) +1\left\{ \begin{array} { l } y = 2 e ^ { - x } \\y = \ln ( x - 2 ) + 1\end{array} \right.{y=2e−xy=ln(x−2) +1​ ​

Solve the system by substitution,if possible.​ {y=7x−219x−y=29\left\{ \begin{array} { l } y = 7 x - 21 \\9 x - y = 29\end{array} \right.{y=7x−219x−y=29​ ​

Solve the system by substitution,if possible.​ {4x+3y=238x+6y=24\left\{ \begin{array} { l } 4 x + 3 y = 23 \\8 x + 6 y = 24\end{array} \right.{4x+3y=238x+6y=24​ ​

​Solve the system of equations by graphing.​ {2x+3y=286x+5y=60\left\{ \begin{array} { l } 2 x + 3 y = 28 \\6 x + 5 y = 60\end{array} \right.{2x+3y=286x+5y=60​ ​ ​

Solve the system by the method of substitution.​ {y=x3−6x2+4y=x2−10x+4\left\{ \begin{array} { l } y = x ^ { 3 } - 6 x ^ { 2 } + 4 \\y = x ^ { 2 } - 10 x + 4\end{array} \right.{y=x3−6x2+4y=x2−10x+4​ ​