Math / Algebra

QuestionA line passes through the points $(-2,5)$ and $(3,10)$ . Write its equation in slope-intercept form.
Write your answer using integers, proper fractions, and improper fractions in simplest form. $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two given points and write it in the form $y = mx + b$ . Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember that the slope-intercept form is $y = mx + b$ , not $y = bx + m$ .

STEP 2

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

STEP 3

Let's define our points.
We have point 1 as $(-2, 5)$ and point 2 as $(3, 10)$ .
This means our first point has $x_1 = -2$ and $y_1 = 5$ , and our second point has $x_2 = 3$ and $y_2 = 10$ .

STEP 4

Now, let's calculate the slope, which is the change in $y$ divided by the change in $x$ .
The formula for slope, usually represented by $m$ , is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Plugging in our values, we get: $m = \frac{10 - 5}{3 - (-2)}$ $m = \frac{5}{3 + 2}$ $m = \frac{5}{5}$ $m = 1$ So, our slope is $m = 1$ !

STEP 5

We know that the y-intercept is the value of $y$ when $x = 0$ .
We can use the slope-intercept form of a linear equation, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
We already know $m = 1$ , and we can use one of our points to find $b$ .
Let's use point 1: $(-2, 5)$ .

STEP 6

Substitute the values into the equation: $5 = 1 \cdot (-2) + b$ $5 = -2 + b$ Now, let's isolate $b$ by adding 2 to both sides of the equation: $5 + 2 = -2 + b + 2$ $7 = b$ So, our y-intercept is $b = 7$ !

STEP 7

We found that the slope $m = 1$ and the y-intercept $b = 7$ .
We can plug these values back into the slope-intercept form equation: $y = mx + b$ $y = 1 \cdot x + 7$ $y = x + 7$

STEP 8

The equation of the line in slope-intercept form is $y = x + 7$ .

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QuestionA line passes through the points $(-2,5)$ and $(3,10)$ . Write its equation in slope-intercept form.
Write your answer using integers, proper fractions, and improper fractions in simplest form. $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two given points and write it in the form $y = mx + b$ . Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember that the slope-intercept form is $y = mx + b$ , not $y = bx + m$ .

STEP 2

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

STEP 3

Let's define our points.
We have point 1 as $(-2, 5)$ and point 2 as $(3, 10)$ .
This means our first point has $x_1 = -2$ and $y_1 = 5$ , and our second point has $x_2 = 3$ and $y_2 = 10$ .

STEP 4

STEP 5

We know that the y-intercept is the value of $y$ when $x = 0$ .
We can use the slope-intercept form of a linear equation, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
We already know $m = 1$ , and we can use one of our points to find $b$ .
Let's use point 1: $(-2, 5)$ .

STEP 6

Substitute the values into the equation: $5 = 1 \cdot (-2) + b$ $5 = -2 + b$ Now, let's isolate $b$ by adding 2 to both sides of the equation: $5 + 2 = -2 + b + 2$ $7 = b$ So, our y-intercept is $b = 7$ !

STEP 7

We found that the slope $m = 1$ and the y-intercept $b = 7$ .
We can plug these values back into the slope-intercept form equation: $y = mx + b$ $y = 1 \cdot x + 7$ $y = x + 7$

STEP 8

The equation of the line in slope-intercept form is $y = x + 7$ .

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QuestionA line passes through the points (−2,5)(-2,5)(−2,5) and (3,10)(3,10)(3,10). Write its equation in slope-intercept form.Write your answer using integers, proper fractions, and improper fractions in simplest form. □\square□

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two given points and write it in the form y=mx+by = mx + by=mx+b. Watch out! Don't mix up the xxx and yyy coordinates!Also, remember that the slope-intercept form is y=mx+by = mx + by=mx+b, not y=bx+my = bx + my=bx+m.

STEP 2

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

STEP 3

Let's **define our points**.We have point 1 as (−2,5)(-2, 5)(−2,5) and point 2 as (3,10)(3, 10)(3,10).This means our first point has x1=−2x_1 = -2x1​=−2 and y1=5y_1 = 5y1​=5, and our second point has x2=3x_2 = 3x2​=3 and y2=10y_2 = 10y2​=10.

STEP 4

STEP 5

STEP 6

**Substitute** the values into the equation: 5=1⋅(−2)+b5 = 1 \cdot (-2) + b5=1⋅(−2)+b 5=−2+b5 = -2 + b5=−2+bNow, let's **isolate** bbb by adding 2 to both sides of the equation: 5+2=−2+b+25 + 2 = -2 + b + 25+2=−2+b+2 7=b7 = b7=bSo, our **y-intercept** is b=7b = 7b=7!

STEP 7

We found that the slope m=1m = 1m=1 and the y-intercept b=7b = 7b=7.We can plug these values back into the **slope-intercept form** equation: y=mx+by = mx + by=mx+b y=1⋅x+7y = 1 \cdot x + 7y=1⋅x+7y=x+7y = x + 7y=x+7

STEP 8

The equation of the line in slope-intercept form is y=x+7y = x + 7y=x+7.

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QuestionA line passes through the points $(-2,5)$ and $(3,10)$ . Write its equation in slope-intercept form.
Write your answer using integers, proper fractions, and improper fractions in simplest form. $\square$

What is this asking? We need to find the equation of a line that goes through two given points and write it in the form $y = mx + b$ . Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember that the slope-intercept form is $y = mx + b$ , not $y = bx + m$ .

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

Let's define our points.
We have point 1 as $(-2, 5)$ and point 2 as $(3, 10)$ .
This means our first point has $x_1 = -2$ and $y_1 = 5$ , and our second point has $x_2 = 3$ and $y_2 = 10$ .

Substitute the values into the equation: $5 = 1 \cdot (-2) + b$ $5 = -2 + b$ Now, let's isolate $b$ by adding 2 to both sides of the equation: $5 + 2 = -2 + b + 2$ $7 = b$ So, our y-intercept is $b = 7$ !

We found that the slope $m = 1$ and the y-intercept $b = 7$ .
We can plug these values back into the slope-intercept form equation: $y = mx + b$ $y = 1 \cdot x + 7$ $y = x + 7$

The equation of the line in slope-intercept form is $y = x + 7$ .