# WORD PROBLEMS ON SIMULTANEOUS EQUATION

A simultaneous equation, as I said before, is one that consists of two-equation made up of two variables.

To solve the word problem on a simultaneous equation, we represent each sentence with a  letter. The accompanying examples illustrate the steps in solving a word problem on a simultaneous equation.

👉This post extends our posts on simultaneous equation👈

Example 1
Four pencils and six books cost €272. Six pencils and five books cost €328. Find the co
st of one pencil and one book.

Let the cost of one pencil be $p$
Let the cost of one book be $b$

From the first sentence
$4p+6b=272$
From the second sentence
$6p+5b=328$

$4p+6b=272$.....eqn 1
$6p+5b=328$.....eqn 2

To eliminate, you will multiply eqn 1 by $6$ and eqn 2 by $4$
$6(4p+6b=272)$
$4(6p+5b=328)$

$24p+36b=1632$
$24p+20b=1312$
Substracting both equation
$24p+36b=1632$
$-(24p+20b=1312)$
$16b=320$

Divide both side by $16$
$\frac{16b}{16}=\frac{320}{16}$
$b=20$

Substituting the value of $b$ in eqn 1
$4p+6(20)=272$
$4p+120=272$
$4p=272-120$
$4p=152$

Divide both side by $4$
$\frac{4p}{4}=\frac{152}{4}$
$p=38$.

One book cost $€20$, one pen cost $€38$

Example 2
The sum of Ada and Sam's ages is 24 years. Six years ago, Ada was three times as old as Sam. How old are they now?

Let Ada age by $x$
Let Sam age be $y$

From the first sentence
$x+y=24$.....eqn 1

From the second sentence
$(x-6)=3(y-6)$
$x-6=3y-18$
$x-3y=-18+6$
$x-3y=-12$.....eqn 2

Substracting eqn 2 from eqn 1
$x+y=24$
$-(x-3y=-12)$
4y=36

Divide both side by $3$
$\frac{4y}{4}=\frac{36}{4}$
$y=9$

Substituting $y$ in eqn 1
$x+9=24$
$x=24-9$
$x=15$

Ada is $1$ while Sam is $15$

Example 3
The sum of the two numbers is $19$. Their difference is $5$, find the two numbers.

Let the first number be $x$
Let the second number be $y$

From the first sentence
$x+y=19$......eqn 1

From the second sentence
$x-y=5$......eqn 2

Substract eqn 2  from eqn 1
$x+y=19$
$-(x-y=5)$
$2y=14$

Divide both side by 2
$\frac{2y}{2}=\frac{14}{2}$
$y=7$

Inserting the value of $y$ in eqn 1
$x+7=19$
$x=19-7$
$x=12$

The first and second numbers are $12$ and $7$ respectively.

Example 4
Dan's age and Josh's age add up to 25 years. Eight years ago, Dan's was twice as old as Josh. How old are they now?

Let dan age be $d$
Let Josh age be $j$

From the first sentence
$d+j=25$......eqn 1

From the second sentence
$(d-8)=2(j-8)$
$d-8=2j-16$
$d-2j=-16+8$
$d-2j=-8$....... eqn 2

Substract eqn 2 from eqn 1
$d+j=25$
$-(d-2j=-8)$
$3j=33$
$\frac{3j}{3}=\frac{33}{3}$
Divide both sides by 3
$j=11$

Substituting the value of j in eqn 1
$d+11=25$
$d=25-11$
$d=14$

Daniel is $14$ while josh's age is $11$.

Example 5
Six books and three bags cost €234. Five books and two bags cost €184. How much does each cost?

Let the cost of the book be $x$
Let the cost of the bag be $y$

From the first sentence
$6x+3y=234$......eqn 1

From the second sentence
$5x+2y=184$.....eqn 2

For elimination sake, we multiply eqn 2 by 3 and eqn 1 by 2
$2(6x+3y=234)$
$12x+6y=468$

$3(5x+2y=184)$15x+6y=552$Subtracting the derived eqn$15x+6y=552-(12x+6y=468)3x=84$Divide both side by$3\frac{3x}{3}=\frac{84}{3}x=28$Substituting$x$in eqn 2$5(28)+2y=184140+2y=1842y=184-1402y=44\frac{2y}{2}=\frac{44}{2}y=22$A book cost €28 while a bag cost €22 Example 6 The average of two numbers is 11. Their difference is$4$. Find the numbers. Let the first number be$x$Let the second number be$y$From the first sentence$\frac{x+y}{2}=11x+y=22$.....eqn 1 From the second sentence$x-y=4$......eqn 2 Substract eqn 2 from eqn 1$x+y=22-(x-y=4)2y=18y=9$Substituting the value of$y$in eqn 2$x-9=4x=4+9x=13$The two numbers are$13$and$9$. Example 7 In ten years times, a father will be twice as old as his son. Ten years ago, he was six times as old as his son. How old are they now? Let father age by$x$Let son age be$y$From the first sentence$x+10=2(y+10)x+10=2y+20x-2y=20-10x-2y=10$.....eqn 1 From the second sentence$x-10=6(y-10)x-10=6y-60x-6y=-60+10x-6y=-50$...... eqn 2 Subtracting eqn 2 from eqn 1$x-2y=10-(x-6y=-50)4y=60$Divide both side by$4\frac{4y}{4}=\frac{60}{4}y=15$Inserting the value of$y$in eqn 1$x-2(15)=10x-30=10x=10+30x=40$The father is$40$, while his som is$15\$.

Finally, we have come to the end of our series simultaneously. As usual, I will be expecting your comments in the comment box.
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