Problem 63 - Algebra 1

Question

Use a graph to determine whether the given three points seem to lie on the same line. If they do, prove algebraically that they lie on the same line and write an equation of the line. $$ (-2,-1),(3,2),(7,5) $$

Step-by-Step Solution

Answer

The three points (-2,-1), (3,2), and (7,5) do lie on the same line. The constant slope between the points is \(\frac{1}{1}\). The equation of the line on which the points lie can be written as \(y - (-1) = 1 *(x - (-2))\), simplified to \(y = x + 1\).

Step 1: Graphical Determination

Initially, plot the three points (-2,-1), (3,2), and (7,5) on a graph. If all the three points lie on the same line, proceed to step 2.

Step 2: Compute Slope

Next, calculate the slope between the points. The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this case, calculate the slope between point 1 and point 2, and point 2 and point 3.

Step 3: Compare Slopes

If the slopes computed in step 2 are equal, that's an indication that the three points lie on the same line. This is because a straight line has a constant slope.

Step 4: Equation of the Line

Once it is confirmed that the points are collinear i.e., they lie on the same line, then the equation of the line that they lie on can be found. Use the slope formula \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is any of the three points. Substituting one set of coordinates and the slope will give the equation of the line

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when analyzing points in the coordinate plane. A slope essentially measures how steep or flat a line is. If you picture a skateboard ramp, you can consider how steep it feels to ride down—a similar feeling applies to slopes on a graph. The mathematical formula for slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula calculates the difference in the y-values divided by the difference in the x-values between two points. The slope tells us how much the y-value (vertical) increases or decreases as the x-value (horizontal) increases by one unit.

• If the slope is positive, the line goes upwards as you move from left to right.
• If the slope is negative, the line goes downwards.
• If the slope is zero, the line is perfectly horizontal.

In our exercise, we calculate the slope between points like \((-2, -1)\) and \((3, 2)\) as well as between \((3, 2)\) and \((7, 5)\). If these slopes are the same, it indicates all these points lie on the same straight line. That's because a single line maintains a consistent slope between any two points on it.

Equation of a Line

Once you've determined that a set of points are collinear (all lying on a straight line), the next step is to find the equation of this line. The formula used is called the point-slope form and is expressed as:
\( y - y_1 = m(x - x_1) \)
This equation uses a known point \((x_1, y_1)\) and the slope \(m\) that you've calculated. Notably, it helps in defining the relationship between the x and y coordinates of any point on the line.

• Choose any of the given points to substitute into the equation.
• Use the consistent slope value already calculated.

For example, using our exercise's points, you could use \( (3, 2) \) as \( (x_1, y_1) \) and substitute the slope found earlier. Solving the equation turns it into the familiar slope-intercept form of a line, \(y = mx + b\), which is often the easiest format for graphing or interpreting the line's behavior over a set of values.

Graphing Points

Graphing points on a coordinate plane is a fundamental skill in algebra. When graphing, each point represents a pair of values \((x, y)\) where \(x\) is the horizontal position and \(y\) is the vertical position. You can think of it like pinpointing a location on a map using grid coordinates.

• Start by plotting the x-value on the horizontal axis.
• From there, move vertically to plot the y-value.
• Mark the point where these values intersect on the grid.

In our exercise, plot each of the points provided: \((-2, -1)\), \((3, 2)\), and \((7, 5)\). Once plotted, if a straight line can pass through all of them, they are likely collinear. Graphing not only helps in visually checking the alignment of points but also aids in confirming algebraic calculations. Ensuring your graph is accurately drawn is key for better visual insights in such tasks.