Math  /  Algebra

QuestionWrite the linear equation that gives the rule for this table.
x | y ---|--- 2 | 1 3 | 2 4 | 3 5 | 4
Write your answer as an equation with yy first, followed by an equals sign. Submit

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that represents the relationship between xx and yy shown in the table. Watch out! Make sure the equation truly reflects the data in the table.
Don't mix up xx and yy values!

STEP 2

1. Find the slope.
2. Find the y-intercept.
3. Write the equation.

STEP 3

Let's **pick two points** from our table to calculate the **slope**.
How about (2,1)(2, 1) and (3,2)(3, 2)?
Sounds good to me!

STEP 4

Remember, the **slope**, often denoted as mm, is the **change in** yy **divided by the change in** xx.
It's like figuring out how steep our line is!

STEP 5

m=change in ychange in x=y2y1x2x1 m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}

STEP 6

Let's plug in our **chosen points**: (x1,y1)=(2,1)(x_1, y_1) = (2, 1) and (x2,y2)=(3,2)(x_2, y_2) = (3, 2).

STEP 7

m=2132=11=1 m = \frac{2 - 1}{3 - 2} = \frac{1}{1} = 1

STEP 8

Awesome! Our **slope** is m=1m = 1.
This tells us that for every **step to the right** (increase in xx), we take **one step up** (increase in yy).

STEP 9

The **y-intercept** is the value of yy when x=0x = 0.
It's where our line crosses the **y-axis**.
We can use the **slope-intercept form** of a linear equation:

STEP 10

y=mx+b y = mx + b

STEP 11

Where mm is our **slope** (which we already found!) and bb is the **y-intercept** we're looking for.

STEP 12

Let's **plug in** one of our points from the table, say (2,1)(2, 1), and our **slope**, m=1m = 1:

STEP 13

1=(1)(2)+b 1 = (1)(2) + b

STEP 14

Now, let's **solve for** bb:

STEP 15

1=2+b 1 = 2 + b 12=b 1 - 2 = b b=1 b = -1

STEP 16

So, our **y-intercept** is 1-1.

STEP 17

Now we have our **slope**, m=1m = 1, and our **y-intercept**, b=1b = -1.
Let's plug them back into the **slope-intercept form**:

STEP 18

y=mx+b y = mx + b y=(1)x+(1) y = (1)x + (-1) y=x1 y = x - 1

STEP 19

y=x1 y = x - 1

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