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Chapter 4: Problem 79

In the following exercises, graph by plotting points. \(x+y=6\)

Short Answer

Expert verified
Plot the points (0, 6), (2, 4), and (4, 2) and draw a line through them.

Step by step solution

01

Understand the Equation

Given the linear equation: \(x + y = 6\). This equation represents a straight line on the coordinate plane.
02

Choose Values for x

Choose at least three different values for \(x\). For example, let鈥檚 choose \(x = 0\), \(x = 2\), and \(x = 4\).
03

Calculate Corresponding y-values

For each chosen \(x\) value, solve the equation for \(y\).When \(x = 0\), \(0 + y = 6\), so \(y = 6\).When \(x = 2\), \(2 + y = 6\), so \(y = 4\).When \(x = 4\), \(4 + y = 6\), so \(y = 2\).
04

Plot the Points

Plot the points on the coordinate plane. The points to plot are: \((0, 6)\), \((2, 4)\), \((4, 2)\).
05

Draw the Line

Draw a straight line through the plotted points. This line represents the equation \(x + y = 6\).

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Key Concepts

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

plotting points
To graph a linear equation like the one given, we first need to find points that lie on the line. We can do this by plotting points.

Start by selecting values for one variable, either x or y. In this exercise, we chose different values for x. Here鈥檚 how to do it:
  • Pick a few values for x.
  • Plug these values into the equation to solve for y.
  • Write down the coordinates (x, y) for each pair of values.
For instance, with the equation \(x + y = 6\), choosing x-values of 0, 2, and 4 gives us the points (0, 6), (2, 4), and (4, 2). Each individually forms the solution to our linear equation.
coordinate plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is divided into four quadrants by the x-axis (horizontal line) and y-axis (vertical line).

When plotting points, we use ordered pairs (x, y) where x represents how far to move horizontally from the origin (0,0) and y represents how far to move vertically.

For example, take the point (2, 4). Here鈥檚 how you plot it:
  • Start at the origin (0,0).
  • Move 2 units to the right to account for x = 2.
  • From there, move 4 units up to account for y = 4.
Once you have these points plotted on the coordinate plane, you can then draw a line through them to represent the linear equation.
solving equations
Solving equations is a key step when graphing linear equations on a coordinate plane. By understanding how to solve for one variable in terms of the other, we can find points to plot.

In our examples, we start with the equation \(x + y = 6\). To find the corresponding y-values for chosen x-values:
  • For \(x = 0\), solve \(0 + y = 6\) to get \(y = 6\).
  • For \(x = 2\), solve \(2 + y = 6\) to get \(y = 4\).
  • For \(x = 4\), solve \(4 + y = 6\) to get \(y = 2\).
This approach ensures that you generate points that satisfy the equation and hence lie on the line represented by the equation. Practicing solving equations like this helps reinforce the fundamentals of algebra and prepares you for more complex problems.

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Most popular questions from this chapter

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