Simultaneous Equations Word Problems 2

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Question 1 of 5

What are the values of the pronumerals

x

$x$ and

y

$y$ ?

$x =$ $x=$ (45) $c m$ $cm$

$y =$ $y=$ (35) $c m$ $cm$

Question 2 of 5

Find the value of the base angles

$Base Angle =$ $\text(Base Angle) =$ (70) $°$ $°$

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First, recall that the base angles of an isosceles triangle are equal

Equate the values of the base angles and simplify

$3 x - 17$ $3x-17$	$=$ $=$	$x + y$ $x+y$
$3 x - 17$ $3x-17$ $- x$ $-x$	$=$ $=$	$x + y$ $x+y$ $- x$ $-x$	Subtract $x$ $x$ from both sides
$2 x - 17$ $2x-17$ $+ 17$ $+17$	$=$ $=$	$y$ $y$ $+ 17$ $+17$	Add $17$ $17$ to both sides
$2 x$ $2x$ $- y$ $-y$	$=$ $=$	$y + 17$ $y+17$ $- y$ $-y$	Subtract $y$ $y$ from both sides
$2 x - y$ $2x-y$	$=$ $=$	$17$ $17$

Next, recall that interior angles of a triangle is equal to

180 °

$180°$

Equate the values of the interior angles to

180 °

$180°$ and simplify

$3 x - 17 + x + y + 40$ $3x-17+x+y+40$	$=$ $=$	$180$ $180$
$4 x + y + 23$ $4x+y+23$	$=$ $=$	$180$ $180$	Combine like terms
$4 x + y + 23$ $4x+y+23$ $- 23$ $-23$	$=$ $=$	$180$ $180$ $- 23$ $-23$	Subtract $23$ $23$ from both sides
$4 x + y$ $4x+y$	$=$ $=$	$157$ $157$	Subtract $y$ $y$ from both sides

Next, write the simultaneous equations being represented in the problem.

$2$ $2$ $x$ $x$ $-$ $-$ $y$ $y$	$=$ $=$	$17$ $17$	First equation
$4$ $4$ $x$ $x$ $+$ $+$ $y$ $y$	$=$ $=$	$157$ $157$	Second equation

Add the two equations.

$2$ $2$ $x$ $x$ $-$ $-$ $y$ $y$	$=$ $=$	$17$ $17$
$4$ $4$ $x$ $x$ $+$ $+$ $y$ $y$	$=$ $=$	$157$ $157$

$6 x$ $6x$	$=$ $=$	$174$ $174$
$x$ $x$	$=$ $=$	$29$ $29$	Divide both sides by $6$ $6$

Solve for $y$ $y$ .

$2$ $2$ $x$ $x$ $-$ $-$ $y$ $y$	$=$ $=$	$17$ $17$
$2$ $2$ $(29)$ $(29)$ $-$ $-$ $y$ $y$	$=$ $=$	$17$ $17$	Substitute $x = 29$ $x=29$
$58 - y$ $58-y$ $- 58$ $-58$	$=$ $=$	$17$ $17$ $- 58$ $-58$	Subtract $58$ $58$ from both sides
$- y$ $-y$ $\times (- 1)$ $times(-1)$	$=$ $=$	$- 41$ $-41$ $\times (- 1)$ $times(-1)$	Multiply both sides by $- 1$ $-1$
$y$ $y$	$=$ $=$	$41$ $41$

Finally, substitute the values of

x

$x$ and

y

$y$ to get the value of the base angles.

First equation

$x$ $x$ $+$ $+$ $y$ $y$	$=$ $=$	$29$ $29$ $+$ $+$ $41$ $41$	Substitute known values
	$=$ $=$	$70 °$ $70°$

Second equation

$3$ $3$ $x$ $x$ $- 17$ $-17$	$=$ $=$	$3$ $3$ $(29)$ $(29)$ $- 17$ $-17$	Substitute known values
	$=$ $=$	$87 - 17$ $87-17$
	$=$ $=$	$70 °$ $70°$

70 °

$70°$

Question 3 of 5

3. Question
A stack of $37$ $37$ coins consisting of $10$ $10$ cents and $50$ $50$ cents adds up to $$ 9.70$ $$9.70$ . How many $10$ $10$ cent and $50$ $50$ cent coins are there?
- $10 cents =$ $10 \text(cents)=$ (22)
  
  $50 cents =$ $50 \text(cents)=$ (15)
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First, let $x$ $x$ be the number of $10$ $10$ cents coins and $y$ $y$ be the number of $50$ $50$ cents coins

Use these values to create a simultaneous equations

$50$ $50$ $x$ $x$ $+ 50$ $+50$ $y$ $y$ $=$ $=$ $970$ $970$ Total value of coins

$x$ $x$ $+$ $+$ $y$ $y$ $=$ $=$ $37$ $37$ Number of coins

Multiply equation $2$ $2$ by $10$ $10$

$($ $($ $x$ $x$ $+$ $+$ $y$ $y$ $)$ $)$ $\times 10$ $xx10$ $=$ $=$ $37$ $37$ $\times 10$ $xx10$

$10$ $10$ $x$ $x$ $+ 10$ $+10$ $y$ $y$ $=$ $=$ $370$ $370$

Subtract the transformed equation from the first equation.

$10$ $10$ $x$ $x$ $+ 50$ $+50$ $y$ $y$ $=$ $=$ $970$ $970$

$10$ $10$ $x$ $x$ $+ 10$ $+10$ $y$ $y$ $=$ $=$ $370$ $370$

$40 y$ $40y$ $=$ $=$ $600$ $600$

$y$ $y$ $=$ $=$ $15$ $15$ Divide both sides by $40$ $40$

Solve for $x$ $x$ .

$x$ $x$ $+$ $+$ $y$ $y$ $=$ $=$ $37$ $37$

$x$ $x$ $+$ $+$ $15$ $15$ $=$ $=$ $37$ $37$ Substitute $y = 15$ $y=15$

$x + 15$ $x+15$ $- 15$ $-15$ $=$ $=$ $37$ $37$ $- 15$ $-15$ Subtract $15$ $15$ from both sides

$x$ $x$ $=$ $=$ $22$ $22$

$x = 22$ $x =22$ , $y = 15$ $y=15$
Question 4 of 5

4. Question
Jason is three times as old as his son Peter. The sum of their ages is $64$ $64$ . What are their ages?
- Peter's age $=$ $=$ (16)
  
  Jason's age $=$ $=$ (48)
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First, let variables to represent the age of Peter and Jason.

$x=$ Peter’s age

$y=$ Jason’s age

Next, write the simultaneous equations being represented in the problem.

$y$ $=$ $3$ $x$ Jason is three times as old as Peter.

$x$ $+$ $y$ $=$ $64$ The sum of their ages is $64$

Substitute Equation $1$ into $2$ and solve for Peter’s age $(x)$ .

$x$ $+$ $y$ $=$ $64$

$x$ $+$ $3x$ $=$ $64$ $y=3x$

$4x$ $=$ $64$

$x$ $=$ $16$ Divide both sides by $4$

Solve for Jason’s age $(y)$ .

$y$ $=$ $3$ $x$

$y$ $=$ $3$ $(16)$ Substitute $x=16$

$y$ $=$ $48$ Simplify

Peter’s age is $16$ while Jason is $48$ years old.
Question 5 of 5

5. Question
Find the value of $x$ and $y$
- $x=$ (20) $°$
  
  $y=$ (30) $°$
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First, recall that supplementary angles are equal to $180°$

Equate the values of the two lower angles to $180°$ and simplify

$3x+2y+3x$ $=$ $180$

$6x+2y$ $=$ $180$ Combine like terms

Next, recall that alternate angles are equal

Equate the alternate angles and simplify

$2y$ $=$ $3x$

Next, write the simultaneous equations being represented in the problem.

$6$ $x$ $+2$ $y$ $=$ $180$ First equation

$2$ $y$ $=$ $3$ $x$ Second equation

Substitute the value of $2y$ into Equation $1$ and solve for $x$ .

$6$ $x$ $+2$ $y$ $=$ $180$

$6$ $x$ $+$ $3x$ $=$ $180$ $2y=3x$

$9x$ $=$ $180$

$x$ $=$ $20°$ Divide both sides by $9$

Solve for $y$ .

$2$ $y$ $=$ $3$ $x$

$2$ $y$ $=$ $3$ $(20)$ Substitute $x=20$

$2y$ $divide2$ $=$ $60$ $divide2$ Divide both sides by $2$

$y$ $=$ $30°$

$x =20°$ , $y=30°$

$5$ $5$ $x$ $x$ $+ 2$ $+2$ $y$ $y$	$=$ $=$	$295$ $295$	Length
$3$ $3$ $x$ $x$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$275$ $275$	Width

$(5$ $(5$ $x$ $x$ $+ 2$ $+2$ $y$ $y$ $)$ $)$ $\times 2$ $xx2$	$=$ $=$	$295$ $295$ $\times 2$ $xx2$
$10$ $10$ $x$ $x$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$590$ $590$

$10$ $10$ $x$ $x$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$590$ $590$
$3$ $3$ $x$ $x$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$275$ $275$

$7 x$ $7x$	$=$ $=$	$315$ $315$
$x$ $x$	$=$ $=$	$45 c m$ $45 cm$	Divide both sides by $7$ $7$

$3$ $3$ $x$ $x$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$275$ $275$
$3$ $3$ $(45)$ $(45)$ $+ 4$ $+4$ $y$ $y$	$=$ $=$	$275$ $275$	Substitute $x = 45$ $x=45$
$135 + 4 y$ $135+4y$ $- 135$ $-135$	$=$ $=$	$275$ $275$ $- 135$ $-135$	Subtract $135$ $135$ from both sides
$4 y$ $4y$	$=$ $=$	$140$ $140$	Divide both sides by $4$ $4$
$y$ $y$	$=$ $=$	$35 c m$ $35 cm$

Simultaneous Equations Word Problems 2

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Quiz summary

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1. Question

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2. Question

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Nice Job!

3. Question

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Well Done!

4. Question

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Fantastic!

5. Question

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Great Work!

Quizzes

$50$ $50$ $x$ $x$ $+ 50$ $+50$ $y$ $y$	$=$ $=$	$970$ $970$	Total value of coins
$x$ $x$ $+$ $+$ $y$ $y$	$=$ $=$	$37$ $37$	Number of coins

$($ $($ $x$ $x$ $+$ $+$ $y$ $y$ $)$ $)$ $\times 10$ $xx10$	$=$ $=$	$37$ $37$ $\times 10$ $xx10$
$10$ $10$ $x$ $x$ $+ 10$ $+10$ $y$ $y$	$=$ $=$	$370$ $370$

$10$ $10$ $x$ $x$ $+ 50$ $+50$ $y$ $y$	$=$ $=$	$970$ $970$
$10$ $10$ $x$ $x$ $+ 10$ $+10$ $y$ $y$	$=$ $=$	$370$ $370$

$40 y$ $40y$	$=$ $=$	$600$ $600$
$y$ $y$	$=$ $=$	$15$ $15$	Divide both sides by $40$ $40$

$y$	$=$	$3$ $x$	Jason is three times as old as Peter.
$x$ $+$ $y$	$=$	$64$	The sum of their ages is $64$

$x$ $+$ $y$	$=$	$64$
$x$ $+$ $3x$	$=$	$64$	$y=3x$
$4x$	$=$	$64$
$x$	$=$	$16$	Divide both sides by $4$

$6$ $x$ $+2$ $y$	$=$	$180$	First equation
$2$ $y$	$=$	$3$ $x$	Second equation

$6$ $x$ $+2$ $y$	$=$	$180$
$6$ $x$ $+$ $3x$	$=$	$180$	$2y=3x$
$9x$	$=$	$180$
$x$	$=$	$20°$	Divide both sides by $9$