Deck 10: Systems of Linear Equations

Question 1

Find the approximate solution by graphing: $5 x-2 y=9$
$-3 x+3 y=9$

Answer 1

$(5,8)$

Question 2

Find the approximate solution by graphing: $2 x+4 y=4$
$3 x-5 y=-5$

Answer 2

$(0,1)$

Question 3

Find the approximate solution by graphing: $x+3 y=-5$
$2 x-y=10$

Answer 3

$(3.6,-2.9)$

Question 4

Solve by any algebraic method: $5 y+x=21$
$3 y-x=3$

Question 5

Solve by any algebraic method: $8 x+7 y=-23$
$3 x+5 y=8$

Question 6

Solve by any algebraic method: $25 x+15 y=5$
$10 x+10 y=10$

Question 7

Solve by any algebraic method: $9 x-5 y=8$
$3 x+4 y=14$

Question 8

Solve by any method: 3x + 2y = 1, x − 3y = −18

Question 9

Solve by any method. Leave your answers in fractional form: $4 x+5 y=2$
$2 x-5 y=0$

Question 10

Solve by any method: $2 x-5 y=38$
$y=3 x-18$

Question 11

Solve by any method: $3 x+4 y=72$
$4 x-3 y=21$

Question 12

Solve by any method: $5 m=6 n-31$
$n=11+2 m$

Question 13

Solve by any method: $3.2 x-1.6 y=7.0$
$1.9 x+2.5 y=6.6$

Question 14

Solve by any method: $-2.5 x-3.4 y=-7.4$
$2.1 x+2.5 y=6.9$

Question 15

Solve by any method: $3.52 x-2.47 y=-0.037$
$1.09 x+1.50 y=6.68$

Question 16

Solve by any method: $2.75 x-3.43 y=-4.144$
$3.30 x+2.50 y=8.59$

Question 17

Solve by any method: $4.2 x+0.91 y=16$
$3.0 x-4.1 y=11$

Question 18

Solve by any method: $7.41 x-3.01 y=2.11$
$6.92 x+9.00 y=1.00$

Question 19

Solve simultaneously: $\frac{x}{3}-\frac{y}{4}=2$
$\frac{x}{2}-\frac{y}{3}=4$

Question 20

Solve simultaneously: $\frac{x}{5}-5 y=15$
$\frac{x}{3}-4 y=12$

Question 21

Solve simultaneously: $\frac{3}{x}-\frac{6}{y}=2$
$\frac{5}{x}+\frac{10}{y}=10$

Question 22

Solve simultaneously. Leave your answers in fractional form: $\frac{x}{3}+\frac{y}{4}=6$
$\frac{x}{5}+\frac{y}{2}=1$

Question 23

Solve simultaneously. Leave your answers in fractional form: $x+\frac{y}{5}=1$
$\frac{x}{4}+\frac{2 y}{5}=1$

Question 24

Solve simultaneously: $\frac{2}{y}-\frac{2}{x}=\frac{5}{2}$
$\frac{4}{y}-\frac{5}{x}=6$

Question 25

Solve simultaneously: $\frac{x}{4}+\frac{y}{5}=18$
$\frac{x}{6}-\frac{y}{3}=-2$

Question 26

Solve simultaneously: $\frac{2 x}{5}+\frac{5 y}{6}=25$
$x-\frac{4 y}{9}=17$

Question 27

Solve simultaneously. Leave your answers in fractional form: $\frac{3 x}{2}+y=4$
$\frac{4 x}{5}+\frac{2 y}{3}=1$

Question 28

Solve simultaneously: $\frac{x}{7}+\frac{y}{24}=\frac{15}{2}$
$\frac{y}{10}+2 x=76$

Question 29

Solve simultaneously: $\frac{10}{x}+\frac{45}{y}=-4$
$\frac{26}{x}+\frac{10}{y}=11$

Question 30

Solve simultaneously. Leave your answers in fractional form: $\frac{2}{x}+\frac{y}{3}=5$
$\frac{1}{x}-y=3$

Question 31

Solve simultaneously. Keep 3 significant digits in your final answers, but do not round until the end:
$\begin{aligned}& \frac{3 x}{7}+\frac{6 y}{11}=\frac{2}{21} \\& \frac{2 x}{5}-\frac{2 y}{3}=\frac{4}{3}\end{aligned}$

Question 32

Solve simultaneously: $\frac{10}{x}+\frac{45}{y}=-4$
$\frac{26}{x}+\frac{10}{y}=11$

Question 33

Solve simultaneously: $x+\frac{2 x-y+1}{9 y}=\frac{-16}{9}$
$\frac{x-3 y+1}{2}-\frac{x-2 y}{3}=1$

Question 34

Solve simultaneously. Leave your answers in fractional form: $\frac{1}{x}+\frac{1}{y}=\frac{3}{5}$
$\frac{2}{x}+\frac{4}{y}=4$

Question 35

Solve for $x$ and $y$ in terms of the other literal quantities: $2 x+3 y=a$
$5 x-y=b$

Question 36

Solve for $x$ and $y$ in terms of the other literal quantities: $a x+b y=c$
$c x-d y=b$

Question 37

Solve for $x$ and $y$ in terms of the other literal quantities: $a x+b y=4$
$a x-d y=2$

Question 38

The sum of two numbers is twenty-one and their difference is fifteen. What are the numbers?

Question 39

Number $A$ divided by number $B$ equals one-fourth. Number $B$ divided by number $A$ equals two times number $A$ . What are the two numbers?

Question 40

There is a fraction such that if 5 is subtracted from the numerator, its value will be $1 / 4$ , and if 7 is added to the denominator, its value will be $1 / 3$ . What is the fraction?

Question 41

The sum of two numbers is 67 . The quotient of the greater divided by the smaller number is 7 with remainder 3. Determine the two numbers.

Question 42

Runner $Y$ leaves the starting line at 12:00. Five minutes later runner $X$ leaves from the same place and catches up to runner $Y$ at 12:07. Four minutes later, runner $X$ is one hundred metres ahead of runner $Y$ . What is the speed of each runner in $\mathrm{m} / \mathrm{s}$ ?

Question 43

If the width of a rectangle is increased by 4 and the length increased by 1 , the area increases by 62 . If instead the width is increased by 2 and the length decreased by 3 , the area decreases by 12 . Find the original dimensions.

Question 44

A person invested $\$ 5,200$ , part in bonds bearing $5.9 \%$ interest and the remainder in bonds bearing $7.6 \%$ interest. Both investments paid the same amount after the same period of time. How much, to the nearest dollar, was invested in each?

Question 45

The sum of the digits of a two-digit number is 12 . When the digits are reversed, the number is increased by 18 . Find the number.

Question 46

A chemist has two acid solutions, one that contains $14.0 \%$ acid and another with $27.0 \%$ acid. How many litres of each must be mixed to make 250 litres of solution containing $17.64 \%$ acid?

Question 47

A lab technician wants to mix some amount of a $15 \%$ acid solution and another amount of a $25 \%$ acid solution so that his resultant mixture is $8 \overline{0} \mathrm{ml}$ of a $22 \%$ acid solution. What volume of the $15 \%$ acid solution and $25 \%$ acid solution is required?

Question 48

Solve for currents $I_{1}$ and $I_{2}$ in the parallel circuit: $100 I_{1}+35 I_{1}-35 I_{2}=12.0$
$40 I_{2}+35 I_{2}-35 I_{1}=18.0$

Question 49

A $10.0 \mathrm{~kg}$ mass and a $19.5 \mathrm{~kg}$ mass are hung from either end of a bar of negligible weight. If the 10.0 $\mathrm{kg}$ is increased to $12.0 \mathrm{~kg}$ , the balance point shifts by $16.2 \mathrm{~cm}$ . How long is the bar?

Question 50

A horizontal beam of negligible weight is simply supported at each end by columns. When a vertical load as placed on a beam $1.00 \mathrm{~m}$ from the left end, the vertical upward reaction is the right column is $62.5 \mathrm{~N}$ . When the load is increased by $40.0 \mathrm{~N}$ , the reaction in the right column increases to $75.0 \mathrm{~N}$ . How long is the beam?

Question 51

A projectile is launched from the ground. Its height (metres) after $t$ seconds is given by $s=k t^{2}+v_{0} t$ where $k$ is a constant, $v_{0}$ is the initial velocity. If the height at $1.0 \mathrm{~s}$ is $8.7 \mathrm{~m}$ and the height at $1.5 \mathrm{~s}$ is $7.625 \mathrm{~m}$ , determine the height at time $2.0 \mathrm{~s}$ .

Question 52

Solve: $2 y+3 x=50$
$x+4 z=30$
$2 z+3 y=20$

Question 53

Solve: $5 x-y=14$
$10 z-y=12$
$x+5 z=20$

Question 54

Solve: $x+z=35$
$x+y=25$
$y+z=32$

Question 55

Solve: $3 x-z=0$
$y+z=2$
$x+y=5$

Question 56

Solve: $x+2 z=5$
$3 x+4 y=2$
$3 y-2 z=3$

Question 57

Solve: $x+2 y-z=4$
$\begin{aligned}& 2 x-y-3 z=-19 \\& x+5 y+2 z=25\end{aligned}$

Question 58

Solve. Leave your answers in fractional form: $3 x+4 y-z=1$
$\begin{array}{r}2 x-2 y+z=0 \\x+y-z=2\end{array}$

Question 59

Solve: $2 x+4 y+z=5$
$x+y+z=6$
$2 x+3 y+z=6$

Question 60

Solve: $0.50 x+1.0 y+1.5 z=7.0$
$1.0 x+0.50 y+1.0 z=5.0$
$1.5 x+2.0 y+1.5 z=1.0$

Question 61

Solve: 0.2x + 0.2y + 0.2z = 7.0
0.2x − 0.4y + 0.6z = 3.0
0.2x - 0.2y + 0.2z = −1.0

Question 62

Solve. Leave your answers in fractional form: $\frac{1}{x}+\frac{2}{z}=7$
$\begin{array}{r}\frac{1}{y}+\frac{1}{z}=\frac{7}{2} \\\frac{4}{x}-\frac{1}{y}+\frac{4}{z}=20\end{array}$

Question 63

Solve. Leave your answers in fractional form: $\frac{x}{3}+\frac{y}{2}+\frac{z}{4}=1$
$\begin{array}{r}2 x+3 y+2 z=4 \\x+\frac{y}{3}+\frac{z}{2}=2\end{array}$

Question 64

Solve: $x+z=5 a$
$\begin{array}{r}y-z=-6 a \\2 x+3 y=-4 a\end{array}$

Question 65

Solve for $x, y$ , and $z: a x+b y=c x y$
$a x-c z=b x z$
$b y+c z=a y z$

Question 66

Solve. Leave your answers in fractional form: $\frac{1}{x}+\frac{1}{y}-\frac{1}{z}=5$
$\begin{aligned}& \frac{1}{x}+\frac{1}{z}=6 \\& \frac{-1}{x}+\frac{3}{y}-\frac{5}{z}=5\end{aligned}$

Question 67

Solve. Leave your answers in fractional form: $\frac{1}{x}+\frac{2}{y}-\frac{2}{z}=6$
$\begin{aligned}& \frac{1}{x}+\frac{1}{y}=7 \\& \frac{2}{y}-\frac{1}{z}=3\end{aligned}$

Question 68

Solve: $\frac{x}{3}-\frac{y}{2}+\frac{z}{6}=-4$
$\begin{gathered}\frac{2 x}{3}+\frac{4 y}{3}-\frac{z}{2}=29 \\\frac{x}{2}+\frac{y}{3}-\frac{z}{6}=11\end{gathered}$

Question 69

Solve: $x+\frac{y}{2}-\frac{z}{3}=21$
$\begin{aligned}& \frac{x}{3}+y-\frac{z}{2}=19 \\& \frac{x}{2}+\frac{y}{3}-z=26\end{aligned}$

Question 70

Solve: $\frac{2 x+5}{3}-\frac{y-1}{2}+z=11$
$\begin{aligned}& \frac{x+6}{4}+y+\frac{z+1}{6}=11 \\& x-\frac{y+2}{3}-\frac{z-1}{2}=-6\end{aligned}$

Question 71

Solve: $\frac{x-1}{y-2}=\frac{4}{5}$
$\begin{aligned}& \frac{x+2}{z-3}=\frac{7}{3} \\& \frac{y-1}{z+2}=\frac{3}{4}\end{aligned}$

Question 72

The sum of the digits of a certain three-digit number is nine. The sum of the hundreds and the tens digits is equal to the ones digit minus one. The number, divided by nine, equals three times the ones digit. What is the number?

Question 73

A certain number has three digits and the sum of those digits 12 . When the number is divided by the number formed by the hundreds and the ones digits, the answer equals 11 . In addition, the sum of the hundreds and the tens digits is 10 bigger than the ones digit. What is the number?

Question 74

The atomic mass of three compounds are known: methane $\left(\mathrm{CH}_{4}\right)$ is 16.0 , glycerol $(\mathrm{C} 3 \mathrm{H} 8 \mathrm{O} 3)$ is 91.9 , and water $(\mathrm{H} 2 \mathrm{O})$ is 18.0. Calculate the atomic mass of $\mathbf{C}, \mathbf{H}$ , and $\mathbf{O}$ by solving this linear system:
$\mathbf{C}+4 \mathbf{H}=16.0$
$\begin{array}{r}3 \mathrm{C}+8 \mathrm{H}+3 \mathrm{O}=91.9 \\2 \mathbf{H}+\mathbf{O}=18.0\end{array}$

Question 75

A projectile is launched from height $h_{o}$ above the ground. Its height (metres) after $t$ seconds is given by $s=k t^{2}+v_{0} t+h_{0}$ where $k$ is a constant, $v_{0}$ is the initial velocity. If the height at $1.0 \mathrm{~s}$ is $7.75 \mathrm{~m}$ and the height at $1.5 \mathrm{~s}$ is $6.625 \mathrm{~m}$ , and the height at $2.0 \mathrm{~s}$ is $5.00 \mathrm{~m}$ , determine the height at time $2.5 \mathrm{~s}$ .