Question

If the line \((3 x-8 y+5)+a(5 x-3 y+10)=0\) is parallel to \(X\) -axis, then a is (1) \(-\frac{8}{3}\) (2) \(-\frac{3}{5}\) (3) \(-2\) (4) \(-\frac{1}{2}\)

Short answer

Answer: a = -3/5

Step-by-step solution

Rewrite the equation in the form y = mx + b

Start by moving all the terms involving y to the left-hand side of the equation: \((3 x-8 y+5)+a(5 x-3 y+10) = 0\) \((-8 - 3a)y = -3x - 5ax + 5 - 10a\) Now divide both sides by \((-8 - 3a)\): \(y = \frac{-3x - 5ax + 5 - 10a}{(-8 - 3a)}\)

Identify the slope of the line

The slope of the line is the coefficient of x in the expression for y: \(y = \frac{-3x - 5ax}{(-8 - 3a)} = \frac{(-3 - 5a)x}{(-8 - 3a)}\) Thus, the slope of the line is \(m = \frac{(-3 - 5a)}{(-8 - 3a)}\)

Make the slope equal to 0

Since the line is parallel to the X-axis, the slope should be 0. Therefore, we get: \(0 = \frac{(-3 - 5a)}{(-8 - 3a)}\) Multiply both sides by \((-8 - 3a)\): \(0 = -3 - 5a\) Now solve for 'a': \(-3 = 5a\) \(a = -\frac{3}{5}\) #a is -\frac{3}{5}# So the correct answer is option (2).

Key concepts

Slope of a Line

When we talk about the slope of a line in mathematics, we are referring to a number that describes both the direction and the steepness of the line. Slope is often represented by the letter 'm' and is calculated by the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. The formula to find the slope is:
\[ m = \frac{\Delta y}{\Delta x} \]
where \(\Delta y\) is the change in the y-coordinates (vertical change) and \(\Delta x\) is the change in the x-coordinates (horizontal change) of two distinct points on the line. A positive slope indicates an upward slant, a negative slope indicates a downward slant, and a zero slope means the line is horizontal. When a line is said to be parallel to the X-axis, it means that the line has a slope of zero, as there is no change in the y-coordinates regardless of the change in x-coordinates.

Linear Equations in Two Variables

Linear equations in two variables are equations that can be written in the form \[ ax + by = c \] where 'a', 'b', and 'c' are constants. These types of equations create straight lines when plotted on a coordinate graph. Each solution to a linear equation is an ordered pair \((x, y)\) that represents a point on that line.

A key property of linear equations is that they have one degree for both 'x' and 'y'. This means that the variables are not squared, cubed, or raised to any higher power. The graph of these equations will always yield a straight line. When we are given a linear equation in a non-standard form, like the one in our exercise, reorganizing it into standard form or slope-intercept form can often make it easier to understand and solve.

Linear Equations Slope-Intercept Form

The slope-intercept form is considered one of the most helpful representations of a line, expressed as \[ y = mx + b \] In this equation, 'm' represents the slope of the line, and 'b' denotes the y-intercept, which is the point where the line crosses the y-axis. This form is extremely useful because it directly provides the slope and the y-intercept, from which we can quickly sketch the graph of the line.

For a line that is parallel to the X-axis, the slope 'm' will be 0, leading to an equation of the form \[ y = b \] This indicates that for every value of 'x', the value of 'y' remains constant, confirming the horizontal nature of the line. In studying our textbook exercise, the understanding of slope-intercept form was instrumental in determining the value of 'a' that resulted in the line having a zero slope.