Problem 37 - Calculus Single Variable

Question

A general linear equation of a line is given. Find the \(x\) -intercept, the \(y\) -intercept, and the slope of the line. $$ 3 x+4 y=2 $$

Step-by-Step Solution

Answer

X-intercept: \((\frac{2}{3}, 0)\); Y-intercept: \((0, \frac{1}{2})\); Slope: \(-\frac{3}{4}\).

Step 1: Identify the Linear Equation

The given linear equation of the line is \(3x + 4y = 2\). To find relevant features like intercepts and slope, we will manipulate this equation.

Step 2: Find the X-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation \(3x + 4y = 2\).Substitute \(y = 0\) into the equation:\[ 3x + 4(0) = 2 \]This simplifies to \(3x = 2\). Solving for \(x\), we get:\[ x = \frac{2}{3} \]So, the \(x\)-intercept is \(\left( \frac{2}{3}, 0 \right)\).

Step 3: Find the Y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation \(3x + 4y = 2\).Substitute \(x = 0\) into the equation:\[ 3(0) + 4y = 2 \]This simplifies to \(4y = 2\). Solving for \(y\), we get:\[ y = \frac{1}{2} \]So, the \(y\)-intercept is \((0, \frac{1}{2})\).

Step 4: Find the Slope of the Line

To find the slope, convert the equation into slope-intercept form, \(y = mx + b\), where \(m\) is the slope.Start with the equation:\[ 3x + 4y = 2 \]Solve for \(y\):\[ 4y = -3x + 2 \]\[ y = -\frac{3}{4}x + \frac{1}{2} \]Now the equation is in the form \(y = mx + b\), where the slope \(m = -\frac{3}{4}\).

Key Concepts

X-intercept

In any linear equation, the **x-intercept** is the point on a graph where the line crosses the x-axis. At this point, the value of \( y \) is always zero. To find the x-intercept for the equation \( 3x + 4y = 2 \), we substitute \( y = 0 \) into the equation. This transforms our equation into \( 3x = 2 \), giving us the solution \( x = \frac{2}{3} \).
Thus, the x-intercept is \( \left( \frac{2}{3}, 0 \right) \), meaning the line crosses the x-axis at this coordinate.

• Remember, setting \( y = 0 \) is crucial to find where the line intersects the x-axis.
• The location of the x-intercept tells us about the horizontal position where the line meets the x-axis.

Y-intercept

The **y-intercept** is where a line crosses the y-axis on a graph, and at this location, \( x \) is always zero. To calculate the y-intercept in the given equation \( 3x + 4y = 2 \), set \( x = 0 \). This simplifies our equation to \( 4y = 2 \), which gives \( y = \frac{1}{2} \) after solving.
Hence, the y-intercept is at \( \left( 0, \frac{1}{2} \right) \). It's the point at which the line intersects the y-axis.

• Setting \( x = 0 \) helps identify where the line meets the y-axis.
• The y-intercept provides information about the vertical position of the line relative to the origin.

Slope

The **slope** of a line describes its steepness and direction. It is expressed as \( m \) in the slope-intercept form of a line, \( y = mx + b \). To determine the slope from the equation \( 3x + 4y = 2 \), we first need to rearrange into slope-intercept form. Rearranging gives us \( y = -\frac{3}{4}x + \frac{1}{2} \). Here, \( m = -\frac{3}{4} \) is the slope.
The negative sign indicates that the line slopes downwards from left to right.

• A negative slope means the line goes downhill as it moves from left to right.
• A positive slope would indicate an upward trend as you move across the graph.
• The numerical value of the slope tells us how steep the line is; larger absolute values mean a steeper line.