Problem 32 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (-3,-5),(1,9) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (-3,-5) and (1,9) is \(y = 3.5x + 5.5\).

Step 1: Calculate the Slope

Use slope formula \((y_2-y_1) / (x_2-x_1)\) to calculate. Substitute the given points (-3,-5) and (1,9) into the formula to get \(m=(9-(-5)) / (1-(-3))=14/4=3.5\). The slope of the line is 3.5.

Step 2: Find the Y-intercept

To find the y-intercept, substitute the slope (m=3.5) and one of the points, say (-3,-5), into the equation \(y=mx+b\) and solve for \(b\). So, \(-5 = 3.5*(-3) + b\), then, \(b=-5-(-10.5)=5.5\). The y-intercept is 5.5.

Step 3: Write the Equation in Slope-Intercept Form

Now that we know the slope and y-intercept, substitute these values into the slope-intercept form equation \(y=mx+b\). So the equation of the line is \(y = 3.5x + 5.5\).

Step 4: Draw the Line

Plot the original points (-3,-5) and (1,9) on the graph. Draw a line passing through these two points, showing that it does indeed follow the above equation. Make sure the line slopes upwards from left to right to indicate a positive slope.

Key Concepts

Slope Calculation

Understanding the slope of a line is critical in grasping the behavior of linear equations. Simply put, the slope represents how steep a line is and the direction in which it tilts. To calculate it, you need two points that lie on the line. With the coordinates of these two points, you can use the slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

For example, given points \( (-3, -5) \) and \( (1, 9) \) the calculation would be \( m = \frac{9 - (-5)}{1 - (-3)} = \frac{14}{4} = 3.5 \). A positive slope like \( 3.5 \) suggests that the line rises from left to right. Remember, a negative slope would indicate a line that falls from left to right, while a slope of zero corresponds to a horizontal line.

Y-intercept Determination

After calculating the slope, the next step is pinpointing where your line crosses the y-axis, known as the y-intercept. It is denoted by \( b \) in the slope-intercept form equation \( y = mx + b \). The y-intercept provides a starting point for drawing your line on the graph.

To find the y-intercept, pick one of your original points and the calculated slope. Insert these into the equation \( y = mx + b \) and solve for \( b \). If we use point \( (-3, -5) \) and slope \( 3.5 \), the equation becomes \( -5 = 3.5(-3) + b \) which simplifies to \( b = -5 + 10.5 = 5.5 \). Thus, the y-intercept of our line is \( 5.5 \), telling us that the line will cross the y-axis at that point.

Graphing Linear Equations

Once we have the slope and y-intercept, graphing the equation \( y = mx + b \) becomes more intuitive. Begin by plotting the y-intercept, \( b \), on the y-axis. From this point, use the slope, \( m \), to determine the angle of the line. In the case of \( m = 3.5 \), we would rise up 3.5 units for every unit we move to the right.

Plot the given points, \( (-3, -5) \) and \( (1, 9) \) on the coordinate system, and draw a straight line through them. This visual representation confirms the accuracy of our equation. For a slope of \( 3.5 \) and y-intercept of \( 5.5 \), the equation \( y = 3.5x + 5.5 \) demonstrates that correct line on the graph. This line should be ascending and straight, reflecting a consistent increase in the value of \( y \) as \( x \) increases.