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Chapter 2: Problem 9

Graph \(2 x+y=4\)

Short Answer

Expert verified
The equation graphed is a straight line passing through the points (2, 0) and (0, 4).

Step by step solution

01

Find the x-intercept

To find the x-intercept, you can set y to 0 and solve the equation for x. This process gives \(2 x + 0 = 4\), from which x equals \(x=2\). Thus, the x-intercept is \(2\).
02

Find the y-intercept

To find the y-intercept, you can set x to 0 and solve the equation for y. This gives \(2 \cdot 0 + y = 4\), leading to y equals \(y=4\). Thus, the y-intercept is \(4\).
03

Draw the graph

Plot the x-intercept (2,0) and the y-intercept (0,4) on a graph. Then draw a straight line through these two points. The line represents the linear equation \(2 x + y = 4\).

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Key Concepts

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

X-Intercept
The x-intercept is a crucial concept when graphing linear equations. It is the point where the graph crosses the x-axis, meaning the y-value of this point is always zero. To find the x-intercept of any linear equation, you set the y variable to zero and solve for x.
Using the example equation from the textbook, \(2x + y = 4\), setting y to zero gives:
  • \(2x + 0 = 4\)
  • \(2x = 4\)
  • \(x = 2\)
So, the point \((2, 0)\) is the x-intercept of this line.
This means our graph will touch the x-axis at the point where x equals 2. Identifying this point helps in drawing a precise linear graph.
Y-Intercept
Like the x-intercept, the y-intercept is an essential point when plotting linear equations. It marks where the line crosses the y-axis, which means the x-value at this point is zero. To discover the y-intercept, set the x variable to zero and solve for y.
Considering the equation \(2x + y = 4\), setting x to zero results in:
  • \(2 \cdot 0 + y = 4\)
  • \(y = 4\)
Thus, the point \((0, 4)\) is the y-intercept.
This provides a pivotal plot point for graphing because it shows exactly where the line meets the y-axis. Knowing both intercepts simplifies drawing an accurate representation of the linear equation.
Plotting Points
Plotting points is the fundamental step towards visualizing a linear equation. After identifying the intercepts, these become the plot points on a coordinate plane. In graphing, every point is written as\((x, y)\), clarifying exactly where it fits on the graph.
To plot effectively, start by marking the x-intercept \((2, 0)\), a point along the horizontal axis. Similarly, plot the y-intercept \((0, 4)\) along the vertical axis.
  • Make sure the points are marked with precision.
  • A straight edge can help in accurate plotting.
Plotting these points correctly forms the basis for drawing a reliable linear line connecting them both. This stage is crucial for the standard visualization of equations.
Linear Equation Graph
Once you have your intercepts plotted, drawing the linear graph becomes easier. A linear equation graph forms a straight line based on the relationship your equation sets. For the equation \(2x + y = 4\), draw a line through the plotted points. This line represents every (x, y) pair that satisfies the equation.
Here鈥檚 how to ensure accuracy:
  • Use a ruler to connect the intercepts \((2,0)\) and \((0,4)\).
  • Extend the line across the graph to understand the full range of the equation.
The final graph demonstrates how the intercepts directly guide the line's slope and direction, creating a visual representation that reflects the equation's behavior. Understanding this baseline process allows you to visualize and solve linear equations effectively.

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Most popular questions from this chapter

Graph the line using the parametric equations $$x=2-3 t, \quad y=1+2 t$$

Find both the \(x\) and \(y\) intercepts of the line \(3 x-2 y=12\).

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (5,-4) and (1,-4) .

Find the slope of the line passing through the following pair of points. (2,3) and (5,9)

Graph the line that passes through the given point and has the given slope. (1,2) and \(m=-3 / 4\)
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