Problem 34
Question
Find an equation for the line that passes through the poiot \(P(2,7)\) and is parallel to the line \(y-2 x: 5=0\)
Step-by-Step Solution
Verified Answer
The line that is parallel to the given line and passes through the point P(2,7) has the equation \(y = 2x + 3\).
1Step 1: Find the Slope of the Given Line
One key property of a line's equation when it is in the form \(y = mx + c\) is that the coefficient of \(x\) is the line's slope. The given line is \(y - 2x -5 = 0\), which rearranges to \(y = 2x + 5\). Therefore, the slope of the given line, m, is 2.
2Step 2: Substitute Slope and Given Point into Equation of Line
Since the line we're looking for will be parallel to the given line, it will also have a slope of 2. Now we'll use the general equation of a line, \(y = mx + c\), and substitute 2 for m, and the given point P(2,7) into this, which gives us \(7 = 2*2 + c\).
3Step 3: Solve for c
Solving the equation \(7 = 2*2 + c\), we find that \(c = 3\).
4Step 4: Write Down the Equation of the Line
Finally, we can write down an equation for the line using our derived values for m and c in the general equation \(y = mx + c\). This gives us \(y = 2x + 3\). This is the equation for the line that is parallel to the given line and passes through the point P(2,7).
Key Concepts
Parallel LinesSlope CalculationPoint-Slope Form
Parallel Lines
Parallel lines have a fascinating mathematical property: they never intersect. This means they share a unique trait when it comes to their slopes. If two lines are parallel, they have identical slopes. This is crucial for solving problems where we need to find a line parallel to another that passes through a specific point.
The exercise you encountered asked for a line that was parallel to a given one. The original line had an equation that could be rearranged to the form \( y = 2x + 5 \). The slope here is 2. Therefore, any line parallel to it will also have a slope of 2.
In summary, the key to finding parallel lines in an equation is ensuring they share the same slope.
The exercise you encountered asked for a line that was parallel to a given one. The original line had an equation that could be rearranged to the form \( y = 2x + 5 \). The slope here is 2. Therefore, any line parallel to it will also have a slope of 2.
In summary, the key to finding parallel lines in an equation is ensuring they share the same slope.
Slope Calculation
The slope of a line is a measure of its steepness, a concept that is central to understanding linear equations. It's often denoted by the letter \( m \). For any linear equation expressed in the form \( y = mx + c \), \( m \) is the slope.
To find the slope of a line from its equation, you need it in slope-intercept form, \( y = mx + c \). In this case, the slope calculation was straightforward because the equation \( y - 2x - 5 = 0 \) could be rearranged to \( y = 2x + 5 \), showing a slope of 2.
Understanding slope helps in identifying the nature of lines—whether they are parallel, perpendicular, or intersecting at various angles.
To find the slope of a line from its equation, you need it in slope-intercept form, \( y = mx + c \). In this case, the slope calculation was straightforward because the equation \( y - 2x - 5 = 0 \) could be rearranged to \( y = 2x + 5 \), showing a slope of 2.
Understanding slope helps in identifying the nature of lines—whether they are parallel, perpendicular, or intersecting at various angles.
Point-Slope Form
The point-slope form of a linear equation is a powerful tool, especially when you know a point on the line and its slope. The formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point, and \( m \) is the slope.
In your problem, you knew the line was parallel to another with a slope of 2, and it had to pass through the point \((2, 7)\). Using this information in point-slope form helps quickly derive the specific line equation.
This approach led you to solve for \( c \) in the general equation \( y = mx + c \). After inserting the point and rearranging, you discovered the line's equation as \( y = 2x + 3 \). Each step in using point-slope form incrementally builds your understanding of how slopes and points define a line completely.
In your problem, you knew the line was parallel to another with a slope of 2, and it had to pass through the point \((2, 7)\). Using this information in point-slope form helps quickly derive the specific line equation.
This approach led you to solve for \( c \) in the general equation \( y = mx + c \). After inserting the point and rearranging, you discovered the line's equation as \( y = 2x + 3 \). Each step in using point-slope form incrementally builds your understanding of how slopes and points define a line completely.
Other exercises in this chapter
Problem 34
Sketch the set on a number line. \([3, \infty)\).
View solution Problem 34
Solve the inequality and express the solution set as an interval or as the union of intervals. $$|5 x-3|
View solution Problem 35
Find \(f\) such that \(f \circ g=F\) given that $$g(x)=\frac{1+x^{2}}{1+x^{4}}, F(x)=\frac{1+x^{4}}{1+x^{2}}$$
View solution Problem 35
Give the domain and range of the function. $$f(x)=\sqrt{x-4}$$
View solution