Problem 25 - Precalculus

Question

Find an equation for the line that is described, and sketch the graph. Write the final answer in the form \(y=m x+b ;\) (a) Passes through (-3,-1) and has slope 4 (b) Passes through \((5 / 2,0)\) and has slope \(1 / 2\) (c) Has \(x\) -intercept 6 and \(y\) -intercept 5 (d) Has \(x\) -intercept -2 and slope \(3 / 4\) (e) Passes through (1,2) and (2,6)

Step-by-Step Solution

Answer

(a) \(y = 4x + 11,\) (b) \(y = \frac{1}{2}x - \frac{5}{4},\) (c) \(y = -\frac{5}{6}x + 5,\) (d) \(y = \frac{3}{4}x + \frac{3}{2},\) (e) \(y = 4x - 2.\)

Step 1: Identify the Slope and a Point

For part (a), the line passes through the point (-3, -1) and has a slope of 4. We can use the point-slope form of the equation of a line to find the equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Step 2: Plug Values into Point-Slope Formula

Using the point slope formula for part (a), we have \( y + 1 = 4(x + 3) \).

Step 3: Arrange the Equation in Slope-Intercept Form

For part (a), distribute and simplify: \( y + 1 = 4x + 12 \) becomes \( y = 4x + 11 \). This is the equation in slope-intercept form \( y = mx + b \).

Step 4: Part (a) Equation: Slope-Intercept Form

The equation of the line for part (a) is \( y = 4x + 11 \). Sketch the line passing through (-3, -1) with a slope of 4.

Step 4: Build Equation for Part (b)

For part (b), the line passes through \((\frac{5}{2}, 0)\) and has a slope of \(\frac{1}{2}\). Using point-slope form: \( y - 0 = \frac{1}{2}(x - \frac{5}{2}) \).

Step 5: Simplify to Slope-Intercept Form for Part (b)

Distribute to get \( y = \frac{1}{2}x - \frac{5}{4} \).

Step 7: Part (b) Equation: Slope-Intercept Form

The equation for part (b) is \( y = \frac{1}{2}x - \frac{5}{4} \). Sketch the line passing through \((\frac{5}{2}, 0)\) with a slope of \(\frac{1}{2}\).

Step 6: Draft Equation for Part (c) Using Intercepts

For part (c), use the given intercepts \(x\text{-intercept} = 6\) and \( y\text{-intercept} = 5 \). The equation is formed as \( \frac{x}{6} + \frac{y}{5} = 1 \).

Step 7: Rearrange to Slope-Intercept Form for Part (c)

Rearrange the equation \( \frac{x}{6} + \frac{y}{5} = 1 \) to get \( y = -\frac{5}{6}x + 5 \).

Step 10: Part (c) Equation: Slope-Intercept Form

The equation of the line for part (c) is \( y = -\frac{5}{6}x + 5 \). Sketch the line with \(x\)-intercept at 6 and \(y\)-intercept at 5.

Step 8: Form Equation for Part (d) Using Intercept and Slope

For part (d), where the \(x\)-intercept is -2 and the slope is \(\frac{3}{4}\), use the point (-2, 0). The point-slope form starts: \( y - 0 = \frac{3}{4}(x + 2) \).

Step 9: Simplify Part (d) to Slope-Intercept Form

Distribute to get \( y = \frac{3}{4}x + \frac{3}{2} \).

Step 13: Part (d) Equation: Slope-Intercept Form

The equation for part (d) is \( y = \frac{3}{4}x + \frac{3}{2} \). Sketch the line with \(x\)-intercept at -2 and slope \(\frac{3}{4}\).

Step 10: Identify Slope Between Two Points for Part (e)

For part (e), calculate the slope from points (1, 2) and (2, 6) using \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{2 - 1} = 4 \).

Step 11: Build Equation Using Calculated Slope for Part (e)

Use point (1, 2) and slope 4 in point-slope form: \( y - 2 = 4(x - 1) \).

Step 12: Simplify Part (e) Equation to Slope-Intercept Form

Distribute and simplify: \( y - 2 = 4x - 4 \) to \( y = 4x - 2 \).

Step 17: Part (e) Equation: Slope-Intercept Form

The equation for part (e) is \( y = 4x - 2 \). Sketch the line passing through points (1, 2) and (2, 6).

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is a popular way to express a line in algebra due to its simplicity and ease of interpretation. It's given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the \( y \)-intercept. This form is handy because it directly gives you the slope and \( y \)-intercept, making it easy to graph.

• The slope \( m \) indicates the steepness of the line. A larger value means a steeper ascent or descent.
• The \( y \)-intercept \( b \) is where the line crosses the \( y \)-axis when \( x = 0 \).

To convert other forms of linear equations into slope-intercept form, isolate \( y \) on one side of the equation. This makes it straightforward to determine the slope and intercept directly from the equation. Practice identifying \( m \) and \( b \) will make graphing linear equations much easier.
When working through problems, such as determining equations from given points or intercepts, rewriting them in slope-intercept form is a crucial step.

Point-slope form

Graphing linear equations often involves using the point-slope form of a line, especially when given a point and a slope. The point-slope form is expressed as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope.

• This form is particularly useful for constructing an equation when you know one point and the slope.
• It provides a direct approach to build a line's equation, then rearrange it into the easier-to-graph slope-intercept form.

For example, if you're given the point \((-3, -1)\) with a slope \( m = 4 \), substitute these into the formula to get \( y + 1 = 4(x + 3) \). To convert this to slope-intercept form, simply solve for \( y \). This transformation makes the equation suitable for graphing and further manipulation.
Understanding how to move between these forms will help you tackle different types of problems in algebra, allowing you to solve and graph equations efficiently.

Linear equation

Linear equations are fundamental in algebra, representing straight lines when graphed on a coordinate plane. The general form of a linear equation is \( Ax + By = C \), but it's often converted into forms like slope-intercept or point-slope for practicality.

• Linear equations involve variables to the first power; hence, their graph is always a straight line.
• They can describe real-world relationships like speed, distance, or cost calculations.

To solve problems involving linear equations, it's important to identify whether you're given a point, slope, intercepts, or other known values. From here, you can choose the appropriate form to work with, such as slope-intercept form when given a \( y \)-intercept, or point-slope form when starting with a point and a slope.
Understanding the properties of linear equations helps in recognizing patterns and relationships in data, making them a powerful tool in both mathematics and real-world applications.