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Chapter 1: Problem 31

Graph each line by hand. Give the \(x\)- and y-intercepts. \(2 x+5 y=10\)

Short Answer

Expert verified
x-intercept: (5, 0); y-intercept: (0, 2)

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\): \[2(0) + 5y = 10 \ 5y = 10 \ y = \frac{10}{5} = 2\] Therefore, the y-intercept is \((0, 2)\).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\):\[2x + 5(0) = 10 \ 2x = 10 \ x = \frac{10}{2} = 5\] Therefore, the x-intercept is \((5, 0)\).
03

Plot the intercepts

On a graph, plot the points \((0, 2)\) which is the y-intercept, and \((5, 0)\) which is the x-intercept.
04

Draw the line

Draw a straight line through the plotted intercepts \((0, 2)\) and \((5, 0)\). This line represents the equation \(2x + 5y = 10\).

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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 line is the point where the line crosses the x-axis. To find the x-intercept, you set the y-coordinate to zero and solve for x in the equation. This is because any point on the x-axis has a y-coordinate of zero. For the equation given, \(2x + 5y = 10\), setting \(y = 0\) results in:
  • \(2x + 5(0) = 10\)
  • \(2x = 10\)
  • \(x = \frac{10}{2} = 5\)
The x-intercept is \((5, 0)\), meaning the line crosses the x-axis at the point where x is 5. Identifying the x-intercept is crucial because it helps plot one of the defining points of the linear graph.
Finding the y-intercept
The y-intercept is where the line intersects the y-axis. You can find this by setting x to zero in the equation and solving for y. Any point on the y-axis has an x-coordinate of zero. In the equation \(2x + 5y = 10\), substituting \(x = 0\) gives:
  • \(2(0) + 5y = 10\)
  • \(5y = 10\)
  • \(y = \frac{10}{5} = 2\)
Thus, the y-intercept is \((0, 2)\), which means the line crosses the y-axis at the point where y is 2. Knowing the y-intercept helps provide another key point necessary for graphing the line.
The equation of a line explained
The equation of a line is a mathematical way to describe a straight line on a graph. It typically takes the form \(Ax + By = C\), also known as the standard form of a linear equation. In our example, we have \(2x + 5y = 10\). Each term in this equation plays a role:
  • \(Ax\) and \(By\) are the variable terms representing the product of coefficients and the x and y variables.
  • \(C\) represents the constant on the right side of the equation.
The equation of a line provides a complete representation of all points that lie on the line. By knowing the x-intercept and y-intercept solutions, you can plot these on a graph and draw a line through them, visually representing this equation. This process helps solidify understanding of how the algebraic form of a line translates into a graphical representation.

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