Problem 33 - Algebra Form and Function

Question

Find an equation for the line parallel to the graph of (a) \(y=3+5 x\) with a \(y\) -intercept of 10 . (b) \(4 x+2 y=6\) with a \(y\) -intercept of 12 . (c) \(y=7 x+2\) and containing the point (3,22) . (d) \(9 x+y=5\) and containing the point (5,15) .

Step-by-Step Solution

Answer

Question: Find the equation of the line parallel to the given line in each of the following cases: a) \(y=3+5 x\) with a \(y\) -intercept of 10. b) \(4 x+2 y=6\) with a \(y\) -intercept of 12. c) \(y=7 x+2\) and containing the point (3,22). d) \(9 x+y=5\) and containing the point (5,15). Answer: a) \(y=5x+10\) b) \(y=-2x+12\) c) \(y=7x+1\) d) \(y=-9x+60\)

Step 1: Identify the slope of the given line

Since the given equation is in the slope-intercept form (y=mx+b), the slope (m) of the line is 5.

Step 2: Use the given y-intercept to find the equation

We know that the y-intercept is 10, so we can use the general form y=mx+b and plug in the values: y=5x+10. The equation for the line parallel to the given line is \(y=5x+10\). (b) \(4 x+2 y=6\) with a \(y\) -intercept of 12.

Step 1: Identify the slope of the given line

First, we need to convert the equation into slope-intercept form. To do this, we will solve for y: \(4x + 2y = 6\) can be rewritten as \(y = -2x + 3\). Hence, the slope (m) of the line is -2.

Step 2: Use the given y-intercept to find the equation

We know that the y-intercept is 12, so we can use the general form y=mx+b and plug in the values: y=-2x+12. The equation for the line parallel to the given line is \(y=-2x+12\). (c) \(y=7 x+2\) and containing the point (3,22).

Step 1: Identify the slope of the given line

Since the given equation is in the slope-intercept form (y=mx+b), the slope (m) of the line is 7.

Step 2: Use the given point to find the equation

We know that the line contains the point (3, 22). We can use the point-slope form (y-y1=m(x-x1)) substituting the point (3,22) and slope 7: y - 22 = 7(x - 3). Now, simplify the equation: y - 22 = 7x - 21. Add 22 to both sides to solve for y: y = 7x + 1. The equation for the line parallel to the given line is \(y=7x+1\). (d) \(9 x+y=5\) and containing the point (5,15).

Step 1: Identify the slope of the given line

First, we need to convert the equation into slope-intercept form. To do this, we will solve for y: \(9x + y = 5\) can be rewritten as \(y = -9x + 5\). Hence, the slope (m) of the line is -9.

Step 2: Use the given point to find the equation

We know that the line contains the point (5, 15). We can use the point-slope form (y-y1=m(x-x1)) substituting the point (5,15) and slope -9: y - 15 = -9(x - 5). Now, simplify the equation: y - 15 = -9x + 45. Add 15 to both sides to solve for y: y = -9x + 60. The equation for the line parallel to the given line is \(y=-9x+60\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward and popular way to express the equation of a line. This form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) indicates the y-intercept, which is the point where the line crosses the y-axis. This form is especially useful because it immediately reveals both the steepness and the direction of the line through its slope. It also provides a clear view of where the line will intersect the y-axis, making it easy to plot.To transform any linear equation into the slope-intercept form, you simply solve for \(y\). For example, consider the equation \(4x + 2y = 6\). By isolating \(y\) on one side, you can rewrite it as \(y = -2x + 3\), where the slope \(m\) is \(-2\) and the y-intercept \(b\) is \(3\). This method helps in identifying whether lines are parallel because parallel lines have equivalent slopes.

Point-Slope Form

The point-slope form is another handy tool for writing the equation of a line. It's particularly useful when you know a point on the line and the slope. The formula is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a known point on the line.This form allows you to easily write the equation of a line using the slope and a specific point. The structure is convenient because it directly incorporates the point's coordinates. For instance, if a line passes through the point \((3, 22)\) with a slope of \(7\), the equation becomes:

• Determine the correct substitution: \(y - 22 = 7(x - 3)\).
• Expand to simplify: \(y - 22 = 7x - 21\).
• Solve for \(y\): Add \(22\) to both sides to yield \(y = 7x + 1\).

The point-slope form is an excellent way to effectively translate between points and slopes, especially useful in finding parallel lines that must pass through specific points.

Y-Intercept

The y-intercept is a fundamental component of a linear equation. It is the point where the line intersects the y-axis, typically expressed as \(b\) in the equation \(y = mx + b\). In practical terms, it's the value of \(y\) when \(x = 0\).Understanding the y-intercept helps in visualizing and graphing a line. It acts as your starting point on the graph from which the slope will dictate the line's direction and steepness.

• In \(y = 5x + 10\), the y-intercept \(b\) is \(10\), so the line crosses the y-axis at \(y = 10\).
• For \(y = -2x + 12\), the y-intercept is \(12\), meaning the line touches the y-axis at \(y = 12\).

Given its importance, ensuring accuracy in identifying the y-intercept is crucial when drawing parallels or establishing the initial position of any line on a graph.