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Chapter 0: Problem 19

Find the \(x\) -intercept(s) and \(y\) -intercepts(s) (if any) of the graphs of the given equations. $$2 x-y=6$$

Short Answer

Expert verified
The x-intercept is (3, 0) and the y-intercept is (0, -6).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) because the x-intercept is where the graph crosses the x-axis. Substitute \( y = 0 \) into the equation \( 2x - y = 6 \). This gives \( 2x - 0 = 6 \), simplifying to \( 2x = 6 \). Solve for \( x \) by dividing both sides by 2: \( x = \frac{6}{2} \). Thus, \( x = 3 \). The x-intercept is \((3, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) because the y-intercept is where the graph crosses the y-axis. Substitute \( x = 0 \) into the equation \( 2x - y = 6 \). This gives \( 2(0) - y = 6 \), which simplifies to \( -y = 6 \). Solve for \( y \) by multiplying both sides by -1: \( y = -6 \). The y-intercept is \((0, -6)\).

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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 is a key point on a graph where the line crosses the x-axis. Here, the value of y is always zero because the point is on the horizontal axis.
To find the x-intercept for the linear equation, you will set the y-value to zero and solve the equation for x. This was demonstrated in our exercise where we have the equation \( 2x - y = 6 \). After substituting \( y = 0 \), we simplify to \( 2x = 6 \) and get \( x = 3 \). Therefore, the x-intercept is the point \((3, 0)\).
  • Set \( y = 0 \).
  • Simplify equation such that \( x \) is isolated.
  • The result gives the x-intercept as a coordinate (x, 0).
This procedure tells where the graph touches the x-axis, providing insights into graph behavior.
Understanding the y-intercept
The y-intercept is where the graph crosses the y-axis, meaning that the x-value is zero because it's on the vertical axis.
To find the y-intercept in a linear equation, you will set \( x = 0 \) and solve for y. In our exercise, substituting \( x = 0 \) into \( 2x - y = 6 \) simplifies to \( -y = 6 \). Solving for \( y \), we get \( y = -6 \), making the y-intercept \((0, -6)\).
  • Set \( x = 0 \).
  • Isolate \( y \).
  • Solution results in the y-intercept as a coordinate (0, y).
Recognizing the y-intercept helps you understand where a line begins or ends at the y-axis and how steep the line is.
What are Linear Equations
Linear equations are mathematical expressions that describe straight lines when plotted on a graph. They have the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constant values.
In these equations, you solve for either x or y by isolating one of the variables. This helps in predicting where the line will meet the axes.
  • They result in straight lines.
  • Both axes intercepts can be easily found.
  • Useful in estimating and relationships between variables.
Linear equations are foundational in algebra, and knowing how to find intercepts plays a critical role in graphing them. They help us determine important features like slope and intersection points on the coordinate plane.

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