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Equation of a line is $3x - 4y + 10 = 0$. Find its
(a) Slope
(b) $x - $ and $y - $ intercepts.

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Last updated date: 04th Mar 2024
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IVSAT 2024
Answer
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Hint: As we know that the above statement is connected to the linear equation in two variables. An equation of the form $px + qy = r$, where $p,q$ and $r$ are real numbers and the variables $p$ and $q$ are not equivalent to zero, is called a linear equation in two variables. The slope intercept form of a linear equation has the following term where the equation is solved for $y$ in terms of $x:y = a + bx$, $b$ is the slope and $a$ is a constant term.

Complete step by step solution:
As per the question we have the equation $3x - 4y + 10 = 0$.
We know that the slope intercept form of the equation is $y = mx + b$, where $m$is the slope and $b$ is the $y$ -intercept value. Let us first write the above question in its general form.
We can write the above equation as $4y = 3x + 10$; by isolating the term $y$, we have $y = \dfrac{3}{4}x + \dfrac{{10}}{4}$.
Now by comparing this from the general form of the straight line, we know that $m$ is the slope, so it gives us $m = \dfrac{3}{4}$, it is the slope of the line.
We know that $y$ intercept is defined as value of $y$ at $x = 0$, so from the equation we have: $y = \dfrac{3}{4} \times 0 + \dfrac{{10}}{4}$.
Upon adding the values it gives us $y = \dfrac{{10}}{4}$=$\dfrac{5}{2}$, it is the $y - $intercept.
Similarly, We know that $x$intercept is defined as value of $x$ at $y = 0$, so by putting this in the equation we have:$0 = \dfrac{3}{4}x + \dfrac{{10}}{4}$.
On further solving we have $0 = \dfrac{{3x + 10}}{4}$. We can take the denominator to the right hand side i.e. $3x = 10$
So it gives us the value $x = - \dfrac{{10}}{3}$.
Hence this is the $x - $ intercept of the equation.

Therefore, slope of the given equation is is $\dfrac{3}{4}$ and x-intercept is $\dfrac{10}{3}$, y-intercept is $\dfrac{5}{2}$.

Note:
We know that the formula of slope intercept form is $y = mx + b$ where $y$ is the “y” coordinate, $m$ is the slope, $x$ is the “x” coordinate and $b$is the ‘y’ intercept. We can use this form of linear equation to draw the graph of the given equation on the “x” and “y” coordinate plane. We should keep in mind that the conversion of the equation of the line to slope intercept form is done by simple manipulation. $Y$intercept of the line is the point where the line cuts the ‘y’ axis and the slope is tan of the angle that is made by the line on the x- axis.