Verified step-by-step solutions for your class notes.
Write a linear equation in two variables in which $a = 3$, $b = 0$ and $c = -\frac{1}{5}$.
Answer: The standard form is $ax + by + c = 0$.
Substituting the given values: $3x + 0y - \frac{1}{5} = 0$.
This can also be written simply as $3x - \frac{1}{5} = 0$, or multiplied by 5 to remove the fraction: $15x - 1 = 0$.
Complete the following table after expressing the given linear equations in standard form.
| Linear equation in two variables | Standard Form | Coefficient of x | Coefficient of y | Constant term |
|---|---|---|---|---|
| $y - 15 = \sqrt{2}x$ | ||||
| $3y - 2x = 0$ | ||||
| $5x = 3y$ | ||||
| $x = 8$ | ||||
| $3y = 1$ |
| Linear equation | Standard Form ($ax+by+c=0$) | Coefficient of x ($a$) | Coefficient of y ($b$) | Constant term ($c$) |
|---|---|---|---|---|
| $y - 15 = \sqrt{2}x$ | $-\sqrt{2}x + y - 15 = 0$ (or $\sqrt{2}x - y + 15 = 0$) | $-\sqrt{2}$ (or $\sqrt{2}$) | $1$ (or $-1$) | $-15$ (or $15$) |
| $3y - 2x = 0$ | $-2x + 3y + 0 = 0$ | $-2$ | $3$ | $0$ |
| $5x = 3y$ | $5x - 3y + 0 = 0$ | $5$ | $-3$ | $0$ |
| $x = 8$ | $1x + 0y - 8 = 0$ | $1$ | $0$ | $-8$ |
| $3y = 1$ | $0x + 3y - 1 = 0$ | $0$ | $3$ | $-1$ |
(i) The cost of a notebook is twice the cost of a pen. Consider the cost of a notebook to be ₹t and that of a pen to be ₹p. Charlie wrote $t = 2p$ whereas Meera wrote $p = 2t$. Which of these two representations is correct?
(ii) In a one-day International Cricket match... two Indian batsmen together scored 176 runs. Manisha expressed this situation as $x + y = 176$, where the number of runs scored by one batsman is x, and the number of runs scored by the other is y. Is this a correct representation?
Answer:
(i) Charlie's representation, $t = 2p$, is correct. Since the notebook ($t$) is twice as expensive as the pen ($p$), you must multiply the cost of the pen by 2 to equal the cost of the notebook.
(ii) Yes, this is a correct representation. The sum of the runs scored by the first batsman ($x$) and the second batsman ($y$) equals the total runs they scored together (176).
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