Problem 38

Question

Find an equation of the line that is parallel to the given line $l$ and passes through the given point $P$. $l: y=-\frac{1}{2} x+4 ; P=(-1,0)$

Step-by-Step Solution

Verified

Answer

The equation of the line is $ y = -\frac{1}{2}x - \frac{1}{2} $.

1Step 1: Identifying the Slope

The first step in solving this problem is to identify the slope of the given line. The given line is $ y = -\frac{1}{2}x + 4 $, which is in the slope-intercept form $ y = mx + b $, where $ m $ is the slope. Therefore, the slope of the line $ l $ is $ -\frac{1}{2} $.

2Step 2: Using the Slope of the Parallel Line

Since parallel lines have the same slope, the line we are looking for will also have a slope of $ -\frac{1}{2} $.

3Step 3: Applying Point-Slope Form

We will use the point-slope form of a line equation, which is $ y - y_1 = m(x - x_1) $. Here, $ (x_1, y_1) = (-1, 0) $ is the point the line passes through, and $ m = -\frac{1}{2} $ is the slope.

4Step 4: Substituting Values into Point-Slope Form

Substitute the values into the point-slope formula: $ y - 0 = -\frac{1}{2}(x + 1) $. This simplifies to $ y = -\frac{1}{2}x - \frac{1}{2} $.

5Step 5: Simplifying the Equation

The equation $ y = -\frac{1}{2}x - \frac{1}{2} $ is the final simplified form of the line parallel to line $ l $ that passes through point $ P $.

Key Concepts

Slope-Intercept FormPoint-Slope FormParallel Lines

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It's quite intuitive once you break it down. This form of a line equation is given by $ y = mx + b $. Here, $ m $ represents the slope of the line. The slope indicates how steep the line is. On the other hand, $ b $ signifies the y-intercept, which is where the line crosses the y-axis.

The slope-intercept form is useful because:

You can quickly identify the slope and y-intercept just by looking at the equation.
It clearly shows how the y-value changes with every increase in the x-value.

Examining the equation $ y = -\frac{1}{2}x + 4 $, we can see that the slope $ m = -\frac{1}{2} $. This means the line decreases by $ 1 $ unit vertically for every $ 2 $ units it moves horizontally to the right. The y-intercept is $ 4 $, indicating that the line crosses the y-axis at $ y = 4 $.

The importance of the slope-intercept form is immense, especially when discussing parallel lines, as they share the same slope!

Point-Slope Form

Another useful way to express the equation of a line is the point-slope form, primarily used when you know one point on the line and the slope. This form is written as $ y - y_1 = m(x - x_1) $. Here, $ (x_1, y_1) $ are the coordinates of a specific point on the line, and $ m $ represents the slope.

The point-slope form is beneficial in the following ways:

It easily incorporates both the slope and a known point on the line.
It's straightforward when you need to construct a line equation from minimal information.

For the problem at hand, after identifying the slope from the line $ l $, we used point $ P = (-1, 0) $. Applying the point-slope form, we substituted $ m = -\frac{1}{2} $, $ x_1 = -1 $, and $ y_1 = 0 $. This gave us $ y - 0 = -\frac{1}{2}(x + 1) $, which is another form of our line equation, just before we simplified it for presentation.

Parallel Lines

Parallel lines are fascinating because they never intersect, no matter how far they are extended. The key reason for this is that parallel lines have the same slope. This means their steepness or their inclination is identical. In a coordinate plane, if two lines have equal slopes, they will continue along their paths without ever meeting.

When we say that a line is parallel to another, like in our exercise, it is crucial because:

The slopes of two parallel lines are identical, which simplifies the process of finding an equation for a line parallel to another given line.
Only the y-intercept might differ, which means they could be vertically shifted versions of each other.

In our exercise, we determined the line parallel to $ l: y = -\frac{1}{2}x + 4 $ through the point $ P = (-1, 0) $. We started by recognizing that both lines must have the same slope, $ m = -\frac{1}{2} $. Using this slope ensures our new line is parallel to line $ l $. This understanding is foundational for constructing parallel lines on a graph.

Problem 38

Other exercises in this chapter

Problem 38

Let $a$ and $b$ be distinct positive numbers different from $1 .$ Show that $\log _{a} x \neq \log _{b} x$ for $x \neq 1$.

View solution

Problem 38

Let $r$ be any rational number and let $f(x)=\sin x+\sin r x$. Show that $f$ is periodic. (Hint: Let $r=m / n$, where $m$ and $n$ are integers.)

View solution

Problem 38

Sketch the graph of the given equation with the help of a suitable translation. Show both the $x$ and $y$ axes and the $X$ and $Y$ axes. $$ y=\sqrt{x+3}

View solution

Problem 38

Solve the inequality. $$ \frac{2 x^{2}-1}{\left(1-x^{2}\right)^{1 / 2}}

View solution