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Chapter 3: Problem 43

In the following exercises, find the intercepts for each equation.. $$ 2 x+5 y=10 $$

Short Answer

Expert verified
The x-intercept is (5, 0) and the y-intercept is (0, 2).

Step by step solution

01

Finding the x-intercept

To find the x-intercept, set y to 0 and solve for x. Start with the equation: 2x + 5y = 10 Substitute y = 0: 2x + 5(0) = 10 This simplifies to: 2x = 10 Solve for x: x = 5
02

Finding the y-intercept

To find the y-intercept, set x to 0 and solve for y. Start with the equation: 2x + 5y = 10 Substitute x = 0: 2(0) + 5y = 10 This simplifies to: 5y = 10 Solve for y: y = 2
03

State the intercepts

The x-intercept is the point where the graph crosses the x-axis, which is at (5, 0). The y-intercept is the point where the graph crosses the y-axis, which is at (0, 2).

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Key Concepts

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

x-intercept calculation
The x-intercept is where a graph crosses the x-axis. To find the x-intercept of a linear equation, set the y-value to zero and solve for x. Let's take the equation given in the exercise:
\(2x + 5y = 10\).

First, substitute \(y = 0\):
\[2x + 5(0) = 10\]
This simplifies to:
\[2x = 10\]
Then, solve for x:
\[x = 5\]

So, the x-intercept is at the point (5, 0). This means that the graph touches the x-axis at the coordinates (5, 0).
Remember:
  • The x-intercept is always in the form (x, 0).
  • Setting y to 0 lets us isolate and solve for x.
  • This gives you the exact point where the graph crosses the x-axis.
y-intercept calculation
The y-intercept is where a graph crosses the y-axis. To find the y-intercept of a linear equation, set the x-value to zero and solve for y. Let's revisit the given equation:
\(2x + 5y = 10\).

First, substitute \(x = 0\):
\[2(0) + 5y = 10\]
This simplifies to:
\[5y = 10\]
Then, solve for y:
\[y = 2\]

So, the y-intercept is at the point (0, 2). This means that the graph touches the y-axis at the coordinates (0, 2).
Important points:
  • The y-intercept is always in the form (0, y).
  • Setting x to 0 helps us isolate and solve for y.
  • This gives you the exact point where the graph crosses the y-axis.
linear equations
Linear equations are equations of the first degree, meaning they graph as straight lines. They can be written in standard form:
\(Ax + By = C\).

Here, A, B, and C are constants, and x and y are variables. Our given equation,
\(2x + 5y = 10\),
is in this form:
\(A=2\),
\(B=5\), and
\(C=10\).
In general:
  • The coefficients (A and B) determine the slope and direction of the line.
  • The constant (C) shifts the line on the graph.
  • Finding intercepts simplifies the graphing process.
By calculating the x and y intercepts, you can easily draw the line on a graph:
  • Plot the x-intercept (5, 0).
  • Plot the y-intercept (0, 2).
  • Draw a straight line through these points.
Understanding linear equations is fundamental in algebra and is often the first step to more complex math topics.

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