Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$25 y=100-50 x$$

Step-by-Step Solution

Answer

The x-intercept is at (2,0), the y-intercept is at (0,4) and line does not pass through the checkpoint (1,1).

Step 1: compute the x intercept

To find the x intercept, set \(y = 0\) in the equation and solve for \(x\). The x intercept is the value of x when \(y = 0\). So we get: \[25(0) = 100 - 50x \] \[0=100-50x \] \[50x=100 \] \[x = 2\] So the x-intercept is (2,0)

Step 2: compute the y intercept

To find the y intercept, set \(x = 0\) in the equation and solve for \(y\). The y intercept represents the value for \(y\) when \(x\) equals to 0. So we get: \[25y=100-50(0)\] \[25y=100\] \[y=4\] So the y-intercept is (0,4)

Step 3: plot the x and y intercepts on the graph

Plot the x-intercept which is located at (2,0) and the y-intercept is at (0,4).

Step 4: Choose a checkpoint

Select a point not on the line as a check point. For instance, select point (1,1). Now substitute these values into the original equations to verify that they satisfy the equality. So we get: \[25(1)=100-50(1)\] \[25=100-50\] \[25=50\] This is false, therefore the point (1,1) does not lie on the line and our line is correct.

Step 5: Drawing the line

Using the intercept points (2,0) and (0,4) and making sure it does not pass through the checkpoint (1, 1), draw a line.

Key Concepts

x-intercept

The x-intercept is a crucial concept in graphing linear equations. It represents the point where a line crosses the x-axis. To locate it, set the y-value in your equation to zero and solve for x. This point has coordinates

• (x, 0)

indicating no vertical change from the x-axis.

In the provided equation, by setting y to zero, we arrive at \[25 \times 0 = 100 - 50x\] which simplifies to \[x = 2.\]Thus, the x-intercept is at the point (2, 0). This means, for this specific linear relationship, the line crosses the x-axis at x equals 2.

Knowing the x-intercept is vital because even with minimal data, it provides one of the foundational points to sketch the line.

y-intercept

Similar to the x-intercept, the y-intercept is where the graph crosses the y-axis. This happens when x equals zero, showing us the value of y at this crossing.

• (0, y)

These are the coordinates indicating the intersection.

For our example, by setting x to zero in the equation, we find: \[25y = 100 - 50 \times 0\] which results in \[y = 4.\]Thus, the y-intercept is the point (0, 4).

This point is particularly significant because it provides an initial starting place for graphing on the y-axis and further aids in describing the line's slope and angle in relation to the axes.

plotting points

Plotting points is a foundational step in graphing equations. It involves locating specific coordinate points on a graph, using them to determine the line's direction and position.

Here's how to plot the points:

• Start by plotting the x-intercept at (2, 0).
• Next, plot the y-intercept at (0, 4).

Once these points are plotted, draw a straight line through them. This line represents your equation on the graph.

Choosing an additional checkpoint confirms accuracy. If a line does not pass through this selected point, like in our example with point (1, 1), it confirms the plotted intercepts' accuracy.

linear equations

Linear equations are mathematical expressions that result in straight lines when plotted on a graph. They typically take the format of \(ax + by = c\), featuring a consistent slope and a constant rate of change between variables x and y.

In our case, the given equation was \(25y = 100 - 50x.\)To recognize it, rearrange terms to get y equation's standard form:\[y = -2x + 4.\]This shows a slope of -2 and a y-intercept of 4.

• Slope: Dictates the line's steepness and angle.
• Intercepts: Where the line crosses the axes.

Understanding the equation's structure aids in effectively plotting and predicting linear relationships.