Linear Relationships in Tables

Ever wondered how to find the slope of a line without drawing a graph? That's what we're going to learn! When you look at a table of values showing x and y coordinates, you can calculate exactly how steep a line is.

In previous lessons, you used similar triangles on graphs to find slope. Remember that slope represents the rate of change between two quantities - how much y changes when x changes by a certain amount.

Learning to calculate slope from tables gives you a powerful tool for analyzing data quickly. You'll soon be able to tell if relationships are proportional (going through the origin) or non-proportional $having a y-intercept other than zero$ just by looking at the numbers.

Quick Tip: The slope formula will save you time! Instead of always graphing points, you'll be able to calculate slope directly from any two coordinate pairs.

Slope Matching

When looking at graphs, you can estimate their slopes by analyzing how steep they appear. The steeper the line, the greater the absolute value of the slope.

For positive slopes like 1/4 and 5/4, the line rises as it moves from left to right. Graphs A, E, and F show this upward trend. The difference between them is how steep they are - 5/4 is steeper than 1/4.

A slope of 0 means the line is perfectly horizontal, like in graph C. This happens when the y-value doesn't change at all as x increases.

Negative slopes like -3 mean the line falls as it moves from left to right. Graphs B and D demonstrate this downward trend. A slope of -3 is quite steep going downward.

Remember This: The sign of the slope tells you the direction - positive slopes go up as x increases, negative slopes go down, and zero slopes stay flat.

Analyzing a Linear Relationship from a Table

Ron earns credits at an arcade when he wins games. By looking at the table showing his games won and credits earned, we can figure out the pattern of how his credits increase.

This relationship is non-proportional because when Ron had won 0 games today $x = 0$, he already had 120 credits $y = 120$. If it were proportional, he would have started with 0 credits.

The ordered pair (0, 120) tells us Ron's starting point - before winning any games today, he already had 120 credits saved up from previous visits.

From the graph, we can find the slope by calculating: (280-200)/(20-10) = 80/10 = 8. This means Ron earns 8 credits for each game he wins. The slope represents his credit earning rate.

Think About It: When calculating slope from a table, be careful to match up the correct x and y values. If you choose random values like Rhonda did, you'll get incorrect answers!

Finding Totals Using Linear Relationships

When we know the pattern of how Ron's credits increase, we can answer important questions about his past and future gaming.

If Ron earns 8 credits per game and started with 120 credits, we can figure out how many games he had won previously to earn those initial credits: 120 ÷ 8 = 15 games. This tells us Ron had already won 15 games before today.

To find how many more games Ron needs to win to reach 500 credits (after his 40th game today), we can use the linear equation: y = 8x + 120. After 40 games today, he has 440 credits. To reach 500, he needs 60 more credits, which means winning 60 ÷ 8 = 7.5 more games (so 8 more games since he can't win a partial game).

The linear relationship y = 8x + 120 completely describes Ron's credit situation: he started with 120 credits, earns 8 credits per game, and after winning 40 games today has 440 total credits.

Helpful Hint: When you have a linear equation in the form y = mx + b, the m value is your slope (rate of change) and b is your y-intercept (starting value).

Calculating Rate of Change from a Table

Finding slope from a table is actually pretty simple! You just need to compare how much y changes relative to how much x changes between any two points.

Here's how you do it:

1. Choose any two y-values and find their difference (vertical change)
2. Find the difference between their corresponding x-values (horizontal change)
3. Divide the vertical change by the horizontal change

For example, using Ron's data, we can pick the points (0, 120) and (25, 320):

• Vertical change: 320 - 120 = 200 credits
• Horizontal change: 25 - 0 = 25 games
• Slope = 200 ÷ 25 = 8 credits per game

This shows that Ron earns 8 credits for each game he wins. The beauty of linear relationships is that this rate stays constant throughout the table.

Pro Tip: You can choose any two points from the table to calculate the slope. If the relationship is truly linear, you'll get the same answer no matter which points you pick!

The Slope Formula

To calculate slope consistently, we use the formula: m = $y₂ - y₁$/$x₂ - x₁$

This formula helps you find the rate of change between any two points (x₁, y₁) and (x₂, y₂). Just be careful to keep your points in the same order when subtracting both coordinates.

When using the formula with data from a table, you can choose any two rows. For example, in Ron's table:

• If we use (0, 120) and (12, 216), the slope is (216 - 120)/(12 - 0) = 96/12 = 8
• If we use (18, 264) and (40, 440), the slope is (440 - 264)/(40 - 18) = 176/22 = 8

Notice that we get the same answer each time - this confirms the relationship is linear!

When drawing arrows to track which values you're using in calculations, make sure you're consistent with your order. Follow the pattern shown in Example 1, where the arrows connect corresponding x and y values.

Watch Out: If you mix up which points are first and second in the formula, you might get a negative slope when it should be positive (or vice versa).

Working with the Slope Formula

Let's see the slope formula in action using Ron's arcade credits data. We'll calculate the slope step by step:

Step 1: Choose two points from the table. Let's use (12, 216) as our first point and (25, 320) as our second point.

Step 2: Label the points with variables from the formula:

• (x₁, y₁) = (12, 216)
• (x₂, y₂) = (25, 320)

Step 3: Plug the values into the slope formula: m = $y₂ - y₁$/$x₂ - x₁$ = (320 - 216)/(25 - 12) = 104/13 = 8

The slope is 8 credits per game, which tells us Ron earns 8 credits every time he wins a game.

If we try different points from the same table, like (40, 440) and (12, 216), we get: m = (440 - 216)/(40 - 12) = 224/28 = 8

We get the same slope! This confirms the relationship is consistently linear.

Connection: Using the slope formula with table values is just like using similar triangles on a graph. Both methods measure the "rise over run" between points - they're just different ways of finding the same thing!

Practice with Linear Relationships in Tables

Now that you know the slope formula, you can calculate the rate of change for any table of values. Remember the formula: m = $y₂ - y₁$/$x₂ - x₁$

For example, with Carnival Ride Tickets:

• Using (4, 9) and (16, 18): m = (18 - 9)/(16 - 4) = 9/12 = 0.75

For the table with negative values:

• Using (0, -2) and (-1, -17): m = (-17 - (-2))/(-1 - 0) = (-17 + 2)/(-1) = -15/-1 = 15

Looking at days and vitamins:

• Using (7, 23) and (18, 9): m = (9 - 23)/(18 - 7) = -14/11 = -1.27

For points (10, 25) and (55, 40):

• m = (40 - 25)/(55 - 10) = 15/45 = 1/3

And for (4, 19) and (24, 3):

• m = (3 - 19)/(24 - 4) = -16/20 = -0.8

None of these relationships are proportional because none of them pass through the point (0, 0). For a relationship to be proportional, when x = 0, y must also equal 0.

Important Insight: You can tell a lot about a relationship just from its slope! A positive slope means y increases as x increases, while a negative slope means y decreases as x increases.

Special Cases: Horizontal and Vertical Lines

Sometimes, linear relationships can be a bit unusual. Let's look at two special cases:

When all y-values in a table are the same (like all equal to 2), the relationship graphs as a horizontal line. The slope of this line is 0 because y doesn't change at all when x changes.

When all x-values in a table are the same (like all equal to 1), the relationship graphs as a vertical line. The slope of this line is undefined because the denominator in our slope formula would be zero: $y₂ - y₁$/(1 - 1) = $y₂ - y₁$/0

These relationships are still linear, but they're special cases:

• Horizontal lines can be written as y = (some constant)
• Vertical lines can be written as x = (some constant)

Visually, a horizontal line runs flat across the graph, while a vertical line runs straight up and down. When you try to calculate the slope of a vertical line, you end up dividing by zero, which is mathematically undefined.

Cool Fact: The slope tells you exactly how a line behaves. A slope of 0 means the line is horizontal, and an undefined slope means the line is vertical!

Determining If a Relationship Is Linear

How do you know if points will form a straight line when plotted? Simple: check if the slope between any two points is always the same.

This is based on an important mathematical principle: If the slope between every ordered pair in a table is constant, then the ordered pairs will form a straight line when graphed.

To test if a relationship is linear:

1. Calculate the slope between different pairs of points
2. If you get the same slope every time, the relationship is linear
3. If the slopes differ, the relationship is not linear

For example, with the table showing x and y values:

• Slope between (4, 13) and (9, 28): (28 - 13)/(9 - 4) = 15/5 = 3
• Slope between (9, 28) and (11, 34): (34 - 28)/(11 - 9) = 6/2 = 3
• Slope between (11, 34) and (16, 47): (47 - 34)/(16 - 11) = 13/5 = 2.6

Since the last slope (2.6) is different from the others (3), these points will not form a straight line when plotted. The relationship is not linear.

Remember: A linear relationship has a constant rate of change. If the rate changes, the relationship isn't linear!