Question

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(4 x-y-6=0\)

Short answer

The slope of the line is 4 and the y-intercept is -6.

Step-by-step solution

Conversion to Slope-intercept Form

To convert the given equation \(4 x - y - 6 = 0\) to the slope-intercept form (\(y = mx + b\)), solve for \(y\). Start by subtracting \(4x\) from both sides: \(-y = -4x + 6\). Then multiply through by -1: \(y = 4x - 6\). This equation is in the form \((y = mx + b)\) and thus, m (slope) = 4 and b (y-intercept) = -6.

Identify Slope and y-intercept

Now that the equation is in the slope-intercept form, it is easy to read off the slope and the y-intercept. The slope (m) is the coefficient of x, which is 4, and the y-intercept (b) is the constant term, which is -6. Thus, the slope of the line is 4 and the y-intercept is -6.

Sketch the Line

The line can be sketched based on the slope and the y-intercept. Start by marking the y-intercept, -6, on the y-axis. The slope of 4 tells you to rise 4 units and run to the right 1 unit from the y-intercept to plot the next point. Draw a line through these two points to sketch the line.

Key concepts

Slope

The slope of a line is a measure of its steepness and direction. It's an important concept in linear equations, revealing how the line moves across a graph.
The slope is often denoted by the letter \(m\) and can be found in the slope-intercept form of the equation \(y = mx + b\). Here, \(m\) shows how much the \(y\)-coordinate increases or decreases for each unit increase in the \(x\)-coordinate.
When we say a line has a slope of 4, like the line described by the equation \(y = 4x - 6\), it indicates that for every one unit you move to the right on the \(x\)-axis, the \(y\)-value will increase by 4 units. This is why the slope is sometimes referred to as "rise over run."
Understanding the concept of slope helps you predict the shape and direction of any line on a graph:

• If the slope is positive, as in our example, the line moves upwards from left to right.
• If the slope is negative, the line moves downwards.
• A slope of zero means the line is flat, running horizontal with no vertical change.
• An undefined slope, often resulting from a vertical line, implies infinite steepness.

Grasping these basics allows you to tackle more complex linear equations confidently.

Y-Intercept

The y-intercept of a line is the point where the line crosses the \(y\)-axis on a graph. It's represented by the variable \(b\) in the slope-intercept form of a linear equation, \(y = mx + b\).
In the context of our equation \(y = 4x - 6\), the \(y\)-intercept is \(-6\). This means that when \(x\) is zero, the line intersects the \(y\)-axis at \(y = -6\).
Understanding the \(y\)-intercept can be particularly useful in graphing a line, as it gives you a precise point to begin your graph.
Here's why knowing the \(y\)-intercept is key:

• It allows you to easily start plotting a line on a graph, marking the exact point where your line meets the \(y\)-axis.
• It gives insight into how shifts in a line change its position relative to the \(y\)-axis.
• The \(y\)-intercept reveals important initial conditions in real-world situations, like starting amounts or initial positions.

By clearly recognizing the \(y\)-intercept, you equip yourself to create accurate linear representations.

Slope-Intercept Form

The slope-intercept form is a straightforward way of writing linear equations. Its formulation is \(y = mx + b\), where:

• \(m\) represents the slope of the line.
• \(b\) is the \(y\)-intercept, the point where the line crosses the \(y\)-axis.

This form provides a simple method for understanding and graphing straight-line equations.
For the linear equation \(4x - y - 6 = 0\), converting it into the slope-intercept form involves isolating \(y\) on one side. This calculation results in \(y = 4x - 6\), making it clear that the slope is 4, and the \(y\)-intercept is \(-6\).
Using the slope-intercept form offers several benefits:

• Provides a clear visual of how the line is positioned and oriented on a graph.
• Simplifies the process of plotting lines, as all key components are readily available.
• Makes it easy to identify how changes in \(m\) or \(b\) will tilt or shift the line.

By mastering slope-intercept form, you gain the power to translate math into visual graphs with ease.