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Solving Problems with Linear Equations - Questions

Question 1
29319

Five more than twice a certain number is one more than the number itself. Find the number

\(x=-4\)

$$\begin{aligned}
\text{Let}\quad x&= \text{the numher}\\
2 x+5 &=x+1 \\
2 x &=x+1-5 \\
2 x-x &=-4 \\
x &=-4
\end{aligned}
$$

Question 2
29320

Tong has $200 in his wallet, consisting of $10 and $5 notes. If he has twice as many $10 notes as $5 notes, how many $5 notes does Tong have?

\(8\)

\begin{align}&\text{Let n be the number of}\ \$5\ \text{notes}\\
&\begin{aligned}
\therefore 2 n \times 10+n \times 5&=200 \\
25 n &=200 \\
n &=200 \div 25 \\
\therefore n &=8
\end{aligned}
\end{align}

Question 3
29321

My mother is 21 years older than me. 12 years ago, she was double my age. How old am I now? 

\(33\)

\begin{align}&\text{Let}\ x=\ \text{my age now.}\\
&\therefore \text { my mothers age now}=x+21 \\
&\begin{aligned}
x+21-12 & =2(x-12) \\
x+9 & =2 x-24 \\
x+33= & 2 x \\
\therefore x & =33
\end{aligned}
\end{align}

Question 4
29322

A fuel tank in a car is 40% full. 14 litres are added and now the tank is 75% full. How much fuel does the tank hold? 

40 litres

\begin{align}&\text{Let}\ V= \text{volume of tank.}\\
&\begin{aligned}
\therefore \quad \frac{2}{5} V+14&=\frac{3}{4}V\\
\frac{3}{4} V-\frac{2}{5} V&=14 \\
\frac{7}{20} V&=14\\
\quad V&=14 \times \frac{20}{7}\\
\quad V&=40 ~\text{litres}
\end{aligned}
\end{align}

Question 5
29323

A cyclist rides at an initial constant speed for 5 hours and then for 4 hours at a speed of 6 km/h greater than the initial speed. If she rides 294 km altogether, what was her initial speed?

30 km/h

$$\begin{aligned}
\text{First distance}\ d_{1}&=5V\quad (V=\text{initial speed})\\
\text{second distance}\ d_{2}&=4(V+6)\\
\therefore \quad 5V+4(V+6) &=294 \\
5V+4V+24 &=294 \\
9V &=270 \\
V &=30 \mathrm{~km} / \mathrm{h}
\end{aligned}
$$