Question

The equation of the line parallel to \(3 \mathrm{x}-2 \mathrm{y}+7=0\) and making an intercept \(-4\) on X-axis is (1) \(3 \mathrm{x}-2 \mathrm{y}+12=0\) (2) \(3 x-2 y-12=0\) (3) \(3 x+2 y-12=0\) (4) \(3 x+2 y+12=0\)

Short answer

Answer: The correct equation is \(3x - 2y + 12 = 0\).

Step-by-step solution

Find the slope of the given line

To find the slope of the given line, first rewrite it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We have the equation \(3x - 2y + 7 = 0\). Solve for \(y\) as follows: $$-2y = -3x + 7$$ $$y = \frac{3}{2}x-\frac{7}{2}$$ Now, we can see that the slope of this line is \(m = \frac{3}{2}\).

Find the equation of the line parallel to the given line and with an x-intercept of -4

Since the required line is parallel to the given line, it will have the same slope, i.e., \(m = \frac{3}{2}\). We are also given that the x-intercept is -4. The x-intercept is the point where the line crosses the x-axis, and at this point, the y-coordinate is 0. So the point of x-intercept is (-4, 0). Now, use the point-slope form of the equation of a line, which is given by: \(y - y_1 = m (x - x_1)\) In this case, the slope \(m = \frac{3}{2}\), and the point is \((-4, 0)\), so we have: \(y - 0 = \frac{3}{2} (x - (-4))\) Simplify the equation: \(y = \frac{3}{2}x + 6\) Rewrite it in the given form (\(ax + by + c = 0\)): \(3x - 2y + 12 = 0\)

Compare with the given options

Comparing our equation with the given options, we can conclude that the equation of the line parallel to \(3x - 2y +7=0\) and making an intercept \(-4\) on the X-axis is: \(3x - 2y + 12 = 0\) (Option 1)

Key concepts

Slope of a Line

The slope of a line is an important concept in geometry that measures the steepness or angle of a line. It is typically represented by the variable \( m \). The slope is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run) between two points on the line.

In the standard equation of a line \( y = mx + b \), \( m \) denotes the slope.To compute the slope of a line given its equation, you first convert it into the slope-intercept form (\( y = mx + b \)). Consider the equation given in the exercise: \( 3x - 2y + 7 = 0 \). First, solve for \( y \):

• Move the term \( 3x \) to the right: \( -2y = -3x + 7 \).
• Next, divide each term by \( -2 \) to isolate \( y \): \( y = \frac{3}{2}x - \frac{7}{2} \).

This shows the slope \( m = \frac{3}{2} \). This slope represents how much \( y \) increases for every unit increase in \( x \).

The slope is essential for understanding how lines behave in relation to the coordinate axes and in determining parallelism of lines since parallel lines have identical slopes.

X-Intercept

The x-intercept of a line is the point where it crosses the x-axis. At this point, the y-coordinate equals zero. Understanding how to identify or utilize an x-intercept can be crucial in writing or analyzing the equations of lines.

For example, to find the x-intercept of a line given by \( 3x - 2y + 7 = 0 \), set \( y = 0 \) and solve for \( x \):

• \( 3x + 7 = 0 \)
• \( 3x = -7 \)
• \( x = -\frac{7}{3} \)

In the exercise, we need a line that has an x-intercept of \(-4\). This means when \( y = 0 \), \( x \) should equal \(-4\).

Knowing the x-intercept helps establish one crucial point on the line, which along with the slope, can be used to determine the equation of a line.

Point-Slope Form

The point-slope form is a way to write the equation of a line using its slope and any point on the line. It can be very useful when you have these two pieces of information and want to derive the line's equation. The point-slope form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

In our exercise, the slope of the parallel line is \( \frac{3}{2} \) and it passes through the point \( (-4, 0) \), thanks to the given x-intercept. By substituting these values into the point-slope form:

• \( y - 0 = \frac{3}{2}(x + 4) \)

Simplifying, this becomes \( y = \frac{3}{2}x + 6 \). The point-slope form is handy because it provides a clear connection between geometric properties of the line and its algebraic representation.

Equation of a Line

The equation of a line is the mathematical description of all the points that form the line. There are several forms to express this equation, including the slope-intercept form and the standard form. In the exercise, we need to find a line's equation parallel to \(3x - 2y + 7 = 0\) with an x-intercept of \(-4\).

Firstly, calculate the slope from the given line's equation to ensure parallelism, which confirms we have \( m = \frac{3}{2} \). Then, using the point-slope form with the given intercept, we derived the line as \( y = \frac{3}{2}x + 6 \).

Rearranging this into the standard form, \( ax + by + c = 0 \), goes like so:

• Multiply everything by 2 to eliminate the fraction: \( 2y = 3x + 12 \).
• Subtract \( 3x \) from both sides to align it to standard form: \( 3x - 2y + 12 = 0 \).

This expression shows the required form, confirming parallelism and the specific x-intercept. Knowing how to convert between different forms of line equations is useful in many contexts such as graphing, analyzing, and solving geometric problems.