Problem 46 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts and the y-intercepts of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=10 x+50 $$

Step-by-Step Solution

Answer

The x-intercept is (-5,0) and y-intercept is (0,50).

Step 1: Find the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). \[ 0 = 10x + 50 \rightarrow x = -5 \] So the x-intercept is (-5, 0)

Step 2: Find the y-intercept

Set \(x = 0\) in the equation and solve for \(y\). \[ y = 10*0 + 50 \rightarrow y = 50 \] So the y-intercept is (0, 50)

Step 3: Graph the equation and label the intercepts

Draw a graph with both the x and y axis. Place marks where your intercepts are. Now draw a straight line through both marks, because it's a linear equation and will, therefore, be a straight line. This line will go through (-5,0) and (0,50). Label these points.

Key Concepts

Graphing

Graphing a linear equation like \( y = 10x + 50 \) involves creating a visual representation of the equation on a coordinate plane. This helps us understand the relationship between \(x\) and \(y\) values. To graph an equation effectively, follow these steps:

• Start by identifying intercepts, which are key points where the line crosses the axes. This makes it easier to draw the line accurately.
• Next, understand that each linear equation represents a straight line. The general form, \(y = mx + b\), includes \(m\) as the slope and \(b\) as the y-intercept.
• Use the intercepts you've found to plot the line. In this case, the x-intercept at \((-5, 0)\) and the y-intercept at \((0, 50)\).
• Draw a straight line through these points, ensuring it extends across the graph to show the full picture of the equation.

Graphing not only helps in visualizing the solution but also provides a confirmation that calculations are correct. It is essential in various fields of math and science.

X-Intercept

The x-intercept of a line is where the line crosses the x-axis. This point has a \(y\) value of zero since it doesn't move up or down from the x-axis.

• To find this, set \(y = 0\) in the equation \(y = 10x + 50\).
• Solving \(0 = 10x + 50\) gives us \(x = -5\).
• Therefore, the x-intercept is \((-5, 0)\).

X-intercepts are vital for understanding the behavior of the equation concerning the x-axis. They show where the line moves horizontally across the plane.

Y-Intercept

The y-intercept is where the line crosses the y-axis, meaning the point where \(x = 0\). It shows the value of \(y\) when the line starts from the vertical axis.

• To find this, substitute \(x = 0\) in the equation \(y = 10x + 50\).
• Calculating this gives \(y = 50\).
• Hence, the y-intercept is \((0, 50)\).

Y-intercepts are useful in determining the initial value of the function at the y-axis. This point reveals how high or low the line starts before it begins moving left or right across the plane.