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Chapter 2: Problem 59

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Then graph the equation. (Lesson \(2-2\) ) $$ y=7 x $$

Short Answer

Expert verified
The x-intercept and y-intercept are both at (0, 0).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). With the equation \(y = 7x\), substituting \(x = 0\) gives \(y = 7(0) = 0\). Therefore, the y-intercept is \((0, 0)\).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). With the equation \(y = 7x\), substituting \(y = 0\) gives \(0 = 7x\). Solving for \(x\) yields \(x = 0\). Therefore, the x-intercept is \((0, 0)\).
03

Graph the equation

Plot the intercept point \((0, 0)\) on the coordinate plane. Since both intercepts are the same, consider another point for a line. For example, substitute \(x = 1\) in the equation \(y = 7x\), resulting in \(y = 7\), which gives another point \((1, 7)\) to plot. Draw the line passing through \((0, 0)\) and \((1, 7)\). This line represents the equation \(y = 7x\).

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Key Concepts

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

Understanding the x-intercept
The x-intercept of a graph is the point where the graph crosses the x-axis of the coordinate plane. To find it, you'll need to set the value of \( y \) to zero in the equation, and then solve for \( x \). This simple substitution lets you see where the line touches the x-axis. For the equation \( y = 7x \), setting \( y = 0 \) leads to \( 0 = 7x \). Solving this equation, you find that \( x = 0 \) is the x-intercept. In this case, the x-intercept is the origin point, tagged as \((0, 0)\). This shows that the line crosses the x-axis exactly at the origin. Remember, every linear equation will have an x-intercept, unless it鈥檚 a vertical line.
Identifying the y-intercept
The y-intercept is just as important as the x-intercept. It identifies where the graph crosses the y-axis. You find it by setting \( x = 0 \) in the equation and solving for \( y \). This particular action translates to finding the point where the line meets the y-axis. Taking the equation \( y = 7x \) as an example, we substitute \( x = 0 \), which results in \( y = 0 \). This means that the y-intercept is at \((0, 0)\). It also means that this point is the starting position on the y-axis. Understanding y-intercepts can make graphing more intuitive, especially when dealing with different types of linear equations.
The basics of graphing linear equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them. Start by identifying the x-intercept and y-intercept, which often help set initial points. For equations like \( y = 7x \), both intercepts occur at \((0, 0)\), making it the only starting point.
  • Pick another convenient point by selecting a value for \( x \).
  • Calculate the corresponding \( y \) value using the equation.
  • In our example, using \( x=1 \) provides \( y=7 \), giving us the point \((1, 7)\).
With these points, draw a straight line through \((0, 0)\) and \((1, 7)\). Linear equations like \( y = 7x \) form straight lines, showing a consistent relationship between \( x \) and \( y \). Graphing helps visually understand how changes in \( x \) influence \( y \) and vice versa.
Exploring the coordinate plane
The coordinate plane is a two-dimensional space defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is determined by a pair of numbers, or coordinates, written as \((x, y)\). The plane is divided into four quadrants, with the origin \((0, 0)\) being the center.
  • The x-axis typically runs left to right, with positive numbers extending right and negative numbers left.
  • The y-axis extends up for positive numbers and down for negative numbers.
When graphing, understanding this plane helps position points accurately. For instance, the point \((1, 7)\) is located one unit to the right of the origin and seven units up. Keeping this in mind aids in the accurate plotting of points and sketching of lines, ensuring the graph reflects the actual equation being plotted.

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Most popular questions from this chapter

Determine whether \((0,0)\) satisfies each inequality. Write \(y>|x|\)

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