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Chapter 10: Problem 29

In Exercises 29 to \(34,\) draw the line described. Through \((3,-2)\) and with \(m=2\)

Short Answer

Expert verified
The line is given by the equation \( y = 2x - 8 \). Draw it by using the y-intercept at \(-8\) and slope \(2\).

Step by step solution

01

Understand the Problem Statement

We are asked to draw a line that passes through the point \((3, -2)\) and has a slope \(m = 2\). The equation of a line we often use for such a situation is the point-slope form, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point the line goes through.
02

Substitute Values into the Point-Slope Formula

Using the point-slope form of the line equation, substitute \(x_1 = 3\), \(y_1 = -2\), and \(m = 2\) into the equation: \[ y - (-2) = 2(x - 3) \]Simplify the equation to make it ready for further processing:\[ y + 2 = 2(x - 3) \].
03

Simplify the Equation

Expand the equation further by applying the distributive property:\[ y + 2 = 2x - 6 \].Next, isolate the variable \(y\) by subtracting 2 from both sides:\[ y = 2x - 6 - 2 \].So the equation simplifies to:\[ y = 2x - 8 \].
04

Draw the Line on a Graph

Now that we have the equation \(y = 2x - 8\), use it to draw the line. Start by plotting the point where the line intersects the y-axis (the y-intercept), which is \(-8\). From this point, use the slope to find another point: rise 2 units up and run 1 unit to the right from the y-intercept to find another point on the line. Connect these points with a straight line.

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Key Concepts

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

Point-Slope Form
In geometry, when we talk about the equation of a line, one very common form is the "point-slope" form. The point-slope form is super useful because it helps us quickly write down the equation of a line if we know two pieces of information: a point the line passes through and the slope of the line.

The formula for the point-slope form is:
  • \( y - y_1 = m(x - x_1) \)
where:
  • \((x_1, y_1)\) is a point on the line, and
  • \(m\) is the slope of the line.
Using this form is straightforward:
  • Plug in the values of the known point into \((x_1, y_1)\).
  • Use the given slope \(m\).
  • Simplify to find the line's equation.
This method is particularly handy because it directly uses the information given to express the line's characteristics clearly.
Equation of a Line
Equations of lines are like the recipes for drawing lines on a graph. In this context, the equation we derived was \( y = 2x - 8 \). This is in the slope-intercept form, where equations are written as \( y = mx + b \).

Here:
  • \(m\) is the slope (how steep the line is).
  • \(b\) is where the line hits, or intersects, the y-axis, also known as the y-intercept.
The equation tells us everything we need to draw the line. You can predict all kinds of points on the line just by plugging in different values for \(x\) and solving for \(y\). It's like a tool that gives you the full picture of how the line behaves across the graph.
Graphing Lines
Graphing lines can be a fun and rewarding task once you understand how the equations work. To graph the line represented by \( y = 2x - 8 \), you simply follow the steps:
  • Start by locating the y-intercept on the y-axis. Here, the y-intercept is \(-8\), so you begin there.
  • Use the slope to find additional points. For every unit you move to the right (positive direction along the x-axis), move 2 units up because the slope \(m = 2\) means rise over run is 2 over 1.
  • Mark these points on the graph.
  • Connect the dots to form a straight line.
This process visually represents the equation and makes it easier to see the relationship between the variables.
Slope
Slope is a core concept in understanding lines in geometry. It tells us how slanted or steep a line is. The slope \(m\) measures how fast \(y\) changes with \(x\).

Visually, slope is seen as:
  • "rise over run," which means how many units you go up or down (rise) for every unit you go right or left (run).
  • A positive slope, like our \(m = 2\), suggests the line rises as we move from left to right.
  • A negative slope would mean the line falls as we move from left to right.
  • If the slope is zero, the line is perfectly horizontal.
  • If the slope is undefined (which happens when the run is zero), the line is vertical.
Slope is crucial because it gives you a quick sense of direction and steepness鈥攊mportant when graphing lines or comparing different linear equations.

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Most popular questions from this chapter

Where \(C\) is a real number, the lines \(3 x+2 y=C\) \(2 x+y=5,\) and \(x-y=4\) are concurrent. Determine the value of \(C\).

Draw an ideally placed figure in the coordinate system; then name the coordinates of each vertex of the figure. a) A trapezoid b) A trapezoid (midpoints of sides are needed)

Where \(m>0, a>0,\) and \(b>0,\) the graph of \(y=m x+b,\) the axes, and the vertical line through \((a, 0)\) determines a trapezoidal region in Quadrant I. Find an expression for the area of this trapezoid in terms of \(a, b,\) and \(m\)

Points \(A\) and \(B\) have symmetry with respect to the origin \(O .\) Find the coordinates of \(B\) if \(A\) is the point: a) \(\quad(3,-4)\) b) \(\quad(0,2)\) c) \((a, 0)\) d) \(\quad(b, c)\)

Points \(A\) and \(B\) have symmetry with respect to the vertical line where \(x=2 .\) Find the coordinates of \(A\) if \(B\) is the point: a) \(\quad(5,1)\) b) \(\quad(0,5)\) c) \((2, a)\) d) \((b, c)\)
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