Problem 34 Graph the line that satisfies ea... [FREE SOLUTION]

Chapter 2: Problem 34

Graph the line that satisfies each set of conditions. passes through \((2,-1),\) perpendicular to graph of \(2 x+3 y=6\)

Short Answer

Expert verified

The line is \(y = \frac{3}{2}x - 4\).

Step by step solution

Find the Slope of the Given Line

Convert the equation of the given line from standard form, \(2x + 3y = 6\), to slope-intercept form, \(y = mx + b\). This helps to identify the line's slope. To do this, solve the equation for \(y\):\[\begin{align*}2x + 3y &= 6\3y &= -2x + 6\y &= -\frac{2}{3}x + 2.\\end{align*}\]Thus, the slope \(m\) of the given line is \(-\frac{2}{3}\).

Calculate the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Given the original slope \(-\frac{2}{3}\), the perpendicular slope is \(\frac{3}{2}\).

Use Point-Slope Form to Write the Equation of the Desired Line

Use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), with point \((2, -1)\) and slope \(\frac{3}{2}\):\[\begin{align*}y + 1 &= \frac{3}{2}(x - 2).\\end{align*}\]This is the equation in point-slope form.

Simplify the Equation into Slope-Intercept Form

Convert the equation to slope-intercept form (\(y = mx + b\)) for easier graphing: \[\begin{align*}y + 1 &= \frac{3}{2}(x - 2)\y + 1 &= \frac{3}{2}x - 3\y &= \frac{3}{2}x - 4.\\end{align*}\]Now the line equation is in slope-intercept form: \(y = \frac{3}{2}x - 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation

Calculating the slope of a line is essential when dealing with linear equations. The slope describes how steep a line is, indicating the change in the y-coordinate per change in x-coordinate. It is represented by the letter \( m \) in equations like the slope-intercept form. To find the slope \( m \), you can convert line equations from standard form to slope-intercept form. This involves isolating the \( y \)-variable on one side.

Take the equation \( 2x + 3y = 6 \).
Solve for \( y \) to convert it to \( y = mx + b \) (slope-intercept form).
The slope \( m \) is then revealed as \( -\frac{2}{3} \).

Knowing the slope is crucial for determining various properties of the line, such as parallels and perpendiculars.

Point-Slope Form

The point-slope form is an incredibly useful formula for writing the equation of a line when you know one point on the line and the slope. It is written as \( y - y_1 = m(x - x_1) \). In this formula:

\((x_1, y_1)\) are the coordinates of a known point on the line.
\(m\) is the slope of the line.

This form is especially handy because it directly incorporates a specific point into the equation. For instance, if you have a line with slope \( \frac{3}{2} \) passing through the point \((2, -1)\), substituting these values gives:\[y + 1 = \frac{3}{2}(x - 2).\]This equation is easy to interpret and guides you toward further manipulation if necessary.

Slope-Intercept Form

The slope-intercept form, given by \( y = mx + b \), is widely used for its simplicity and clarity. Here:

\( m \) is the slope of the line, showing how y changes with x.
\( b \) is the y-intercept, the point where the line crosses the y-axis.

Converting to this form is often the final step in solving line problems since it reveals both the slope and position clearly. Starting from a point-slope form like \( y + 1 = \frac{3}{2}(x - 2) \), you can simplify to:\[y = \frac{3}{2}x - 4.\]This reveals the line's slope as \( \frac{3}{2} \) and the y-intercept as \(-4\), providing a complete picture of the line's behavior on a graph.

Line Equations

Understanding line equations is foundational in geometry and algebra. These equations are mathematical statements that represent a line's path on a two-dimensional plane. Different forms provide different insights:

The **standard form** \( Ax + By = C \) reaches simple solutions but doesn't clearly reveal slope or intercepts.
The **slope-intercept form** \( y = mx + b \) directly shows the slope and y-intercept, making graphing straightforward.
The **point-slope form** \( y - y_1 = m(x - x_1) \) is intuitive for building an equation from known data points and slopes.

Each form has its advantages. Understanding how to transition between them, like converting point-slope to slope-intercept, is key to mastering the concepts of line equations.

91影视

Graph the line that satisfies each set of conditions. passes through \((2,-1),\) perpendicular to graph of \(2 x+3 y=6\)

Short Answer

Step by step solution

Find the Slope of the Given Line

Calculate the Slope of the Perpendicular Line

Use Point-Slope Form to Write the Equation of the Desired Line

Simplify the Equation into Slope-Intercept Form

Key Concepts

Slope Calculation

Point-Slope Form

Slope-Intercept Form

Line Equations

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