Solved on Jan 18, 2024

Find equivalent linear equations given $14x + 7y = 21$ . Options: $y = 2x + 3$ , $y + 1 = -2(x - 2)$ , $y = -2x + 3$ , $y - 1 = 2(x + 2)$ , $y - 3 = -2x$ .

STEP 1

Assumptions
1. The given equation of the line is $14x + 7y = 21$ .
2. We need to determine which of the provided equations are equivalent to the given equation.
3. To be equivalent, the equations must represent the same line, which means they must have the same slope and y-intercept when written in slope-intercept form ( $y = mx + b$ ).

STEP 2

First, we need to convert the given equation of the line into slope-intercept form to identify its slope and y-intercept.
$14x + 7y = 21$

STEP 3

Isolate $y$ on one side of the equation to get the slope-intercept form.
$7y = -14x + 21$

STEP 4

Divide every term by 7 to solve for $y$ .
$y = -2x + 3$

STEP 5

Now that we have the slope-intercept form of the given line, we can compare its slope and y-intercept with the provided equations. The slope is $-2$ and the y-intercept is $3$ .

STEP 6

Compare the first equation $y = 2x + 3$ with the slope-intercept form of the given line.

STEP 7

Notice that the slope of $y = 2x + 3$ is $2$ , which is not equal to $-2$ , and therefore, this equation is not equivalent to the given line.

STEP 8

Compare the second equation $y + 1 = -2(x - 2)$ with the slope-intercept form of the given line.

STEP 9

Distribute the $-2$ across the parentheses.
$y + 1 = -2x + 4$

STEP 10

Subtract $1$ from both sides to solve for $y$ .
$y = -2x + 3$

STEP 11

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 12

Compare the third equation $y = -2x + 3$ with the slope-intercept form of the given line.

STEP 13

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 14

Compare the fourth equation $y - 1 = 2(x + 2)$ with the slope-intercept form of the given line.

STEP 15

Distribute the $2$ across the parentheses.
$y - 1 = 2x + 4$

STEP 16

Add $1$ to both sides to solve for $y$ .
$y = 2x + 5$

STEP 17

Notice that the slope of $y = 2x + 5$ is $2$ , which is not equal to $-2$ , and therefore, this equation is not equivalent to the given line.

STEP 18

Compare the fifth equation $y - 3 = -2x$ with the slope-intercept form of the given line.

STEP 19

Add $3$ to both sides to solve for $y$ .
$y = -2x + 3$

STEP 20

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 21

After comparing all the provided equations, we can conclude that the equivalent equations to the given line $14x + 7y = 21$ are:
- $y + 1 = -2(x - 2)$ - $y = -2x + 3$ - $y - 3 = -2x$
The equations that are not equivalent are:
- $y = 2x + 3$ - $y - 1 = 2(x + 2)$

Was this helpful?

Find equivalent linear equations given 14x+7y=2114x + 7y = 2114x+7y=21. Options: y=2x+3y = 2x + 3y=2x+3, y+1=−2(x−2)y + 1 = -2(x - 2)y+1=−2(x−2), y=−2x+3y = -2x + 3y=−2x+3, y−1=2(x+2)y - 1 = 2(x + 2)y−1=2(x+2), y−3=−2xy - 3 = -2xy−3=−2x.

STEP 1

STEP 2

First, we need to convert the given equation of the line into slope-intercept form to identify its slope and y-intercept.14x+7y=2114x + 7y = 2114x+7y=21

STEP 3

Isolate yyy on one side of the equation to get the slope-intercept form.7y=−14x+217y = -14x + 217y=−14x+21

STEP 4

Divide every term by 7 to solve for yyy.y=−2x+3y = -2x + 3y=−2x+3

STEP 5

Now that we have the slope-intercept form of the given line, we can compare its slope and y-intercept with the provided equations. The slope is −2-2−2 and the y-intercept is 333.

STEP 6

Compare the first equation y=2x+3y = 2x + 3y=2x+3 with the slope-intercept form of the given line.

STEP 7

Notice that the slope of y=2x+3y = 2x + 3y=2x+3 is 222, which is not equal to −2-2−2, and therefore, this equation is not equivalent to the given line.

STEP 8

Compare the second equation y+1=−2(x−2)y + 1 = -2(x - 2)y+1=−2(x−2) with the slope-intercept form of the given line.

STEP 9

Distribute the −2-2−2 across the parentheses.y+1=−2x+4y + 1 = -2x + 4y+1=−2x+4

STEP 10

Subtract 111 from both sides to solve for yyy.y=−2x+3y = -2x + 3y=−2x+3

STEP 11

Notice that the slope is −2-2−2 and the y-intercept is 333, which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 12

Compare the third equation y=−2x+3y = -2x + 3y=−2x+3 with the slope-intercept form of the given line.

STEP 13

Notice that the slope is −2-2−2 and the y-intercept is 333, which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 14

Compare the fourth equation y−1=2(x+2)y - 1 = 2(x + 2)y−1=2(x+2) with the slope-intercept form of the given line.

STEP 15

Distribute the 222 across the parentheses.y−1=2x+4y - 1 = 2x + 4y−1=2x+4

STEP 16

Add 111 to both sides to solve for yyy.y=2x+5y = 2x + 5y=2x+5

STEP 17

Notice that the slope of y=2x+5y = 2x + 5y=2x+5 is 222, which is not equal to −2-2−2, and therefore, this equation is not equivalent to the given line.

STEP 18

Compare the fifth equation y−3=−2xy - 3 = -2xy−3=−2x with the slope-intercept form of the given line.

STEP 19

Add 333 to both sides to solve for yyy.y=−2x+3y = -2x + 3y=−2x+3

STEP 20

Notice that the slope is −2-2−2 and the y-intercept is 333, which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

STEP 21

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find equivalent linear equations given $14x + 7y = 21$ . Options: $y = 2x + 3$ , $y + 1 = -2(x - 2)$ , $y = -2x + 3$ , $y - 1 = 2(x + 2)$ , $y - 3 = -2x$ .

First, we need to convert the given equation of the line into slope-intercept form to identify its slope and y-intercept.
$14x + 7y = 21$

Isolate $y$ on one side of the equation to get the slope-intercept form.
$7y = -14x + 21$

Divide every term by 7 to solve for $y$ .
$y = -2x + 3$

Now that we have the slope-intercept form of the given line, we can compare its slope and y-intercept with the provided equations. The slope is $-2$ and the y-intercept is $3$ .

Compare the first equation $y = 2x + 3$ with the slope-intercept form of the given line.

Notice that the slope of $y = 2x + 3$ is $2$ , which is not equal to $-2$ , and therefore, this equation is not equivalent to the given line.

Compare the second equation $y + 1 = -2(x - 2)$ with the slope-intercept form of the given line.

Distribute the $-2$ across the parentheses.
$y + 1 = -2x + 4$

Subtract $1$ from both sides to solve for $y$ .
$y = -2x + 3$

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

Compare the third equation $y = -2x + 3$ with the slope-intercept form of the given line.

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.

Compare the fourth equation $y - 1 = 2(x + 2)$ with the slope-intercept form of the given line.

Distribute the $2$ across the parentheses.
$y - 1 = 2x + 4$

Add $1$ to both sides to solve for $y$ .
$y = 2x + 5$

Notice that the slope of $y = 2x + 5$ is $2$ , which is not equal to $-2$ , and therefore, this equation is not equivalent to the given line.

Compare the fifth equation $y - 3 = -2x$ with the slope-intercept form of the given line.

Add $3$ to both sides to solve for $y$ .
$y = -2x + 3$

Notice that the slope is $-2$ and the y-intercept is $3$ , which matches the slope-intercept form of the given line. Therefore, this equation is equivalent to the given line.