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Chapter 4: Problem 178

Find the intercepts for each equation. \(y=5 x\)

Short Answer

Expert verified
The intercepts are both at (0, 0).

Step by step solution

01

Identify the Equation

The given equation is a linear equation: \(y = 5x\).
02

Find the Y-Intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). When \(x = 0\), \(y = 5 \times 0 = 0\). Thus, the y-intercept is \( (0, 0) \).
03

Find the X-Intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). When \(y = 0\), \(0 = 5x\). Solving for \(x\), we get \(x = 0\). Thus, the x-intercept is \( (0, 0) \).

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Key Concepts

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

linear equations
Linear equations form a fundamental part of algebra. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. For example, the equation provided in the exercise, \(y = 5x\), is a linear equation. The general form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Linear equations are called such because the variables appear to the first power, and the graph of the equation is a straight line. This simplicity is what helps make linear equations one of the first and most understandable topics in algebra.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. At this point, the value of \(x\) is always zero. For example, in the linear equation \(y = 5x\), we find the y-intercept by setting \(x = 0\). When we substitute \(x = 0\) into \(y = 5x\), we get:
\(y = 5 \cdot 0 = 0\).
This tells us that the y-intercept is \((0, 0)\).
Why is the y-intercept useful?
  • It helps in quickly graphing a line.
  • It provides an initial value point when \(x\) is zero.
Knowing how to find the y-intercept is a crucial step in understanding linear equations and graphing.
x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \(y\) is always zero. For instance, in the equation provided \(y = 5x\), to find the x-intercept, we set \(y = 0\). Substituting \(y = 0\) into the equation, we get:
\(0 = 5x\).
Solving for \(x\), we find that \(x = 0\). Thus, the x-intercept is \((0, 0)\).
Why is the x-intercept important?
  • It helps in graphing the line more accurately.
  • It gives a specific point where the line meets the x-axis.
Understanding the x-intercept is equally vital as knowing the y-intercept for complete comprehension of linear equations.

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