Question

The equation of a line passing through \((4,3)\) and the sum of whose intercepts is \(-1\) is........ (a) \((\mathrm{x} / 2)+(\mathrm{y} / 3)=1,(\mathrm{x} / 2)+(\mathrm{y} / 1)]=1\) (b) \((\mathrm{x} / 2)+(\mathrm{y} / 3)=-1,[\mathrm{x} /(-2)]+(\mathrm{y} / 1)=1\) (c) \((\mathrm{x} / 2)+(\mathrm{y} / 3)=-1,\\{\mathrm{x} /(-2)\\}+(\mathrm{y} / 1)=-1\) (d) \((\mathrm{x} / 2)-(\mathrm{y} / 3)=1,\\{\mathrm{x} /(-2)\\}+(\mathrm{y} / 1)=1\)

Short answer

The correct equation of the line is \((x/2)-(y/3)=1\), which is option (d).

Step-by-step solution

Understanding the Statement

The given statement the equation of a line passing through \((4,3)\) implies that when we place x=4 and y=3 in the line equation, this should satisfy the line equation. In other words, if our line equation \(x/a+y/b=n\), then if we place x=4 and y=3, the left side should be equal to n.

Applying the condition of the line passing through a point

Apply the values given in the equations from (a) to (d), and discard the choices that do not satisfy this condition. (a) Substituting x=4 and y=3 in \((x/2)+(y/3)=1\\), we get \(4/2+3/3=1 \\) or 2+1=1, which is not correct. Hence option (a) is not correct. (b) Substituting x=4 and y=3 in \((x/2)+(y/3)=-1 \\) we get \(4/2+3/3=-1 \\) or 2+1=-1, which is not correct. Hence option (b) is not correct. (c) Substituting x=4 and y=3 in \((x/2)+(y/3)=-1 \\), we get \(4/2+3/3=-1 \\) or 2+1=-1, which is incorrect. Hence, option (c) is incorrect. (d) Substituting x=4 and y=3 in \((x/2)-(y/3)=1 \\), we get \(4/2-3/3=1 \\) or 2-1=1, which is correct. So, only option (d) fulfills the condition for the line passing through point \((4,3)\).

Checking the condition of sum of intercepts

In the correct equation from step 2 \((x/2)-(y/3)=1\), the intercepts are \(a=2\) and \(b=-3\). Their sum is \(2 + -3 = -1\), which meets the provided condition of sum of intercepts being \(-1\). Therefore, the equation of the line is \( (x/2)-(y/3)=1 \) or option (d) is the correct answer.

Key concepts

Intercepts

Intercepts are crucial in understanding the geometric positioning of a line on a coordinate plane. They are points where the line crosses the axes. Specifically, the intercept on the x-axis is where the line crosses the x-axis ( x-intercept ) and the intercept on the y-axis is where it crosses the y-axis ( y-intercept ).

• An x-intercept is found when the value of y is zero in the equation of the line. It tells you at what point the line will touch the x-axis.
• A y-intercept is found when x is zero, showing where the line will intersect with the y-axis.

In this exercise, it was given that the sum of the intercepts is -1. This means: \[ a + b = -1 \]where \(a\) is the x-intercept, and \(b\) is the y-intercept. In the correct equation \((x/2)-(y/3)=1\), the intercepts are found as follows:

• The x-intercept \(a\) is 2 when y is 0.
• The y-intercept \(b\) is -3 when x is 0.

Adding these values confirms the sum is -1, satisfying the condition.

Linear Equations

A linear equation in two variables can graphically represent a straight line in a coordinate plane. These equations are typically written as \(ax + by = c\) or rearranged into different forms such as \((x/a) + (y/b) = 1\), where a and b represent the intercepts.

• The general form of a linear equation is often expressed as \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants.
• Another common form is the slope-intercept form, expressed as \(y = mx + c\) where \(m\) is the slope of the line and \(c\) is the y-intercept.

In the context of this problem, the line equation \((x/2)-(y/3)=1\) talks about intercepts in fractions which indicate intercepts directly as 2 for x and -3 for y. Such forms make it straightforward to analyze various properties of the line like how it meets the axes.

Coordinate Geometry

Coordinate geometry provides a systematic way of describing the positions and relationships between points on a plane. By using an x and y-axis, each point has a specific coordinate usually written in the form \((x, y)\).

• This system allows for a clear representation of geometric figures and phenomena through algebraic equations.
• Each point on a line can be plugged into a linear equation to verify if it lies on that line, similar to point \((4, 3)\) in this problem.

In this exercise, coordinate geometry helps in confirming that the line equation \((x/2)-(y/3)=1\) passes through the point \((4, 3)\). By substituting these coordinates into the equation and verifying if it holds true, it confirms the relationship between the point and line which is a classic application of the concepts of coordinate geometry. This branch of mathematics makes it easier to design, measure, and analyze geometric shapes by reducing them into numeric formulas.