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Chapter 3: Problem 43

Find the intercepts of the graph of \(2 x-5 y=20\) [ 2.4]

Short Answer

Expert verified
The intercepts are (10, 0) and (0, -4).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation is: \(2x - 5(0) = 20\) This simplifies to: \(2x = 20\) Divide both sides by 2: \(x = 10\) So, the x-intercept is (10, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The equation is: \(2(0) - 5y = 20\) This simplifies to: \(-5y = 20\) Divide both sides by -5: \(y = -4\) So, the y-intercept is (0, -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 of a graph is the point where the line crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept for the equation given, you need to set y to 0 and solve for x.

Following the example equation:

Step 1: Set y to 0: \(2x - 5(0) = 20\)

Step 2: Simplify the equation: \(2x = 20\)

Step 3: Solve for x: \(x = 10\)

Thus, the x-intercept is at the point (10, 0).

Remember:
  • The x-intercept will always have y set to zero.
  • It represents the point where the graph of the equation crosses the x-axis.
  • This concept is crucial for graphing linear equations.
y-intercept
The y-intercept of a graph is the point where the line crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept, you need to set x to 0 and solve for y.

Following the example equation:

Step 1: Set x to 0: \(2(0) - 5y = 20\)

Step 2: Simplify the equation: \(-5y = 20\)

Step 3: Solve for y: \(y = -4\)

Thus, the y-intercept is at the point (0, -4).

Important points to remember:
  • The y-intercept will always have x set to zero.
  • It represents the point where the graph of the equation crosses the y-axis.
  • Knowing the y-intercept helps in graphing linear equations accurately.
linear equations
Linear equations are equations of the first degree, meaning they have the highest power of the variable as one. The general form of a linear equation is \(ax + by = c\). These equations graph as straight lines.

In the given exercise, the linear equation is: \(2x - 5y = 20\).

Key characteristics:
  • Linear equations can have one or more variables.
  • The graph of a linear equation in two variables (x and y) is a straight line.
  • To graph a linear equation, you often need to find the x-intercept and the y-intercept.


Solving a linear equation involves finding values for the variables that make the equation true. By finding the intercepts, you are finding specific points that satisfy the equation and help in graphing the line accurately.

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