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a and
b are the intercepts made by
a line on the co-ordinate axes. If
3a=b and the line passes through
(1,3), then the equation of the line is

$a$ and $b$ are the intercepts made by a line on the co-ordinate axes. If $3a=b$ and the line passes through $(1,3)$ , then the equation of the line is $\newline$ A) $x+3y =10$ $\newline$ B) $3x+y =6$ $\newline$ C) $x-3y+8 =0$ $\newline$ D) $3x-2+3 =0$

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Transformations of linear functions

Full solution

Q.

a

and

b

are the intercepts made by a line on the co-ordinate axes. If

3a=b

and the line passes through

(1,3)

, then the equation of the line is

\newline

A)

x+3y =10

\newline

B)

3x+y =6

\newline

C)

x-3y+8 =0

\newline

D)

3x-2+3 =0

Identify Relationship: Identify the relationship between the intercepts $a$ and $b$ . Given that $3a = b$ , we can express $b$ in terms of $a$ . Calculation: $b = 3a$
Determine Equation: Determine the equation of the line using intercept form. $\newline$ The intercept form of a line's equation is $\frac{x}{a} + \frac{y}{b} = 1$ . $\newline$ Since we know $b = 3a$ , we can substitute $b$ in the equation. $\newline$ Calculation: $\frac{x}{a} + \frac{y}{(3a)} = 1$
Use Given Point: Use the given point $(1,3)$ to find the value of $a$ . $\newline$ Substitute $x = 1$ and $y = 3$ into the equation $\frac{x}{a} + \frac{y}{3a} = 1$ . $\newline$ Calculation: $\frac{1}{a} + \frac{3}{3a} = 1$
Simplify Equation: Simplify the equation to solve for $a$ . Combine the terms on the left side of the equation. Calculation: $\frac{1}{a} + \frac{1}{a} = 1$ Calculation: $\frac{2}{a} = 1$
Solve for a: Solve for a. Multiply both sides by $a$ to isolate $a$ . Calculation: $2 = a$
Find Value of b: Find the value of b using the relationship $b = 3a$ . $\newline$ Substitute $a = 2$ into $b = 3a$ . $\newline$ Calculation: $b = 3 \times 2$ $\newline$ Calculation: $b = 6$
Write Final Equation: Write the final equation of the line using the values of $a$ and $b$ . Substitute $a = 2$ and $b = 6$ into the intercept form equation $\frac{x}{a} + \frac{y}{b} = 1$ . Calculation: $\frac{x}{2} + \frac{y}{6} = 1$
Multiply for Standard Form: Multiply through by the least common multiple of the denominators to get the standard form of the equation. $\newline$ Calculation: $3(\frac{x}{2}) + (\frac{y}{6}) = 3$ $\newline$ Calculation: $\frac{3x}{2} + \frac{y}{6} = 3$ $\newline$ Calculation: $6(\frac{3x}{2} + \frac{y}{6}) = 6(3)$ $\newline$ Calculation: $9x + y = 18$

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