Q1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.

Solution:
Let cost of pen be \(y\)
and cost of Note Book =\(x\)
\(\because\) Cost of Note book is twice that of pen
\(\therefore\quad x=2y\)
Linear Equation in two variable is \[\begin{align}\implies x&=2y\\ \implies x-2y&=0 \end{align} \]

Express the following linear equations in the form \(ax + by + c = 0\) and indicate the values of \(a\), \(b\) and \(c\) in each case:

1. \(2x + 3y = 9.3\bar{5}\)
2. \(x-\frac{y}{5}-10=0\)
3. \(–2x + 3y = 6\)
4. \( x = 3y\)
5. \(2x = –5y\)
6. \(3x + 2 = 0\)
7. \( y – 2 = 0\)
8. \( 5 = 2x\)

Solution:

• i. \[2x + 3y = 9.3\bar5\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} 2x + 3y &= 9.3\bar5\\ 2x+3y-9.35&=0\tag{2} \end{align} \] Comparing coefficient og Eqn (1) and (2) \[ \begin{align} a&=2\\ b&=3\\ c&=-9.3\bar5 \end{align} \]
• ii.\[x-\frac{y}{5}-10=0\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} x-\frac{y}{5}-10&=0\\ \end{align} \] Multiplying both side by 5 \[ \begin{align} 5 \times \left(x - \frac{y}{5} - 10 \right) &= 0\times5 \\ \implies 5x - y - 50 &= 0 \tag{2} \\ \end{align} \] Comparing coefficients of Eqn (1) and (2): \[ \begin{align}a &= 5 \notag \\ b &= -1 \notag \\ c &= -50 \notag \end{align} \]
• iii.\[–2x + 3y = 6\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} –2x + 3y &= 6\\ -2x+3y-6&=0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=-2\\ b&=3\\ c&=-6 \end{aligned} \]
• iv \[x = 3y\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} x &= 3y\\ x-3y+0&=0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=1\\ b&=-3\\ c&=0 \end{aligned} \]
• v. \[2x = –5y\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} 2x &= –5y\\ 2x+5y+0&=0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=2\\ b&=5\\ c&=0 \end{aligned} \]
• vi. \[3x + 2 = 0\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} 3x + 2 &= 0\\ 3x + + 0\cdot y +2 &= 0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=3\\ b&=0\\ c&=2 \end{aligned} \]
• vii. \[y – 2 = 0\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} y – 2 &= 0\\ 0\cdot x +y-2&= 0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=0\\ b&=1\\ c&=-2 \end{aligned} \]
• \[5 = 2x\] General Form \[ax+by+c=0\tag{1}\] writing given equation in general form
  \[ \begin{align} 5 &= 2x\\ 5-2x&=0\\ -2x+0\cdot y +5&=0\tag{2}\end{align}\] Comparing coefficients of Eqn (1) and (2) \[ \begin{aligned} a&=-2\\ b&=0\\ c&=5 \end{aligned} \]