Problem 51 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(3 x+4 y-1=0\)

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{4}\) and the y-intercept is \((0, \frac{1}{4})\).

Step 1: Identify the coefficients

The given line equation is in the general form: \( ax + by + c = 0 \). For the equation \( 3x + 4y - 1 = 0 \), the coefficients are \( a = 3 \), \( b = 4 \), and \( c = -1 \).

Step 2: Convert to Slope-Intercept Form

To find the slope, rewrite the equation in the form \( y = mx + b \) (Slope-Intercept Form). Start by solving for \( y \):\[ 4y = -3x + 1 \] Divide every term by 4:\[ y = -\frac{3}{4}x + \frac{1}{4} \]

Step 3: Identify the Slope

In the equation \( y = mx + b \), \( m \) represents the slope. Here, \( m = -\frac{3}{4} \). Thus, the slope of the line is \(-\frac{3}{4}\).

Step 4: Identify the y-intercept

In the equation \( y = mx + b \), \( b \) represents the y-intercept. Here, \( b = \frac{1}{4} \). Thus, the y-intercept is \((0, \frac{1}{4})\).

Step 5: Graph the Line

Start by plotting the y-intercept \((0, \frac{1}{4})\). From this point, use the slope \(-\frac{3}{4}\) to find another point. A slope of \(-\frac{3}{4}\) means go down 3 units and right 4 units. From \((0, \frac{1}{4})\), moving 4 units right and 3 units down gives the point \((4, -\frac{11}{4})\). Connect these points to draw the line.

Key Concepts

Understanding the Slope

The slope of a line is a number that measures its steepness and direction. In simple terms, it tells us how slanted a line is, and whether it goes up or down as we move along it from left to right.
If a line tilts upwards, its slope is positive. If it tilts downward, the slope is negative. When a line is perfectly flat, it has a slope of zero.

• The slope is represented by the letter "m" in the equation of a line.
• It is a ratio of the change in y (vertical) to the change in x (horizontal).

For the given equation, once in slope-intercept form, we find the slope to be \(-\frac{3}{4}\). This value means that for every 4 units we move to the right along the x-axis, the line goes 3 units down.
Visualizing this can be helpful: picture standing on a hill that is not very steep, tilting downwards. That's what a negative slope does.

Recognizing the Y-intercept

The y-intercept is where the line crosses the y-axis. In mathematical terms, it is the value of \(y\) when \(x\) is 0.
Identifying the y-intercept is crucial because it gives us a point where the line starts on the y-axis.

• In the equation of a line, the y-intercept is represented by the letter "b" in the form \(y = mx + b\).
• The y-intercept is expressed as a point (0, b), where \(x = 0\).

For the equation in question, we have found that the y-intercept is \((0, \frac{1}{4})\). Here's a simple visualization: On a graph, the line touches the y-axis slightly above zero.

Graphing Lines with Slope and Y-intercept

Graphing lines can be intuitive once you know the slope and y-intercept.
These two characteristics provide all the information you need to draw the line accurately.

• Begin by plotting the y-intercept on a graph. For our exercise, it's at \((0, \frac{1}{4})\).
• Then use the slope to find the next point. A slope of \(-\frac{3}{4}\) means you move 3 units down for every 4 units you move to the right.

After plotting these two points,
draw a straight line through them.
This line represents all possible points that satisfy the equation. Each point on this line is a solution to the equation \(3x + 4y - 1 = 0\).
Visualize it like a string extending infinitely in both directions, passing exactly through these plotted points.