Solved on Jan 19, 2024

Determine which equation is in standard form: a. $x+3=-5y$ , b. $5-y=x$ , c. $y=3x+6$ , d. $-8x+3y=12$ .

STEP 1

Assumptions
1. Standard form for a linear equation is $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is non-negative.
2. We are given four options and we need to identify which one is in standard form.

STEP 2

Examine option a: $x+3=-5y$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 3

Rearrange option a to move the $y$ term to the same side as the $x$ term.
$x + 5y = -3$

STEP 4

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $x + 5y = -3$ , all coefficients are integers and the coefficient of $x$ is positive (1). However, the equation is not in standard form because the constant term is negative.

STEP 5

Examine option b: $5-y=x$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 6

Rearrange option b to move the $x$ term to the same side as the $y$ term.
$-x - y = -5$

STEP 7

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-x - y = -5$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

STEP 8

Examine option c: $y=3x+6$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 9

Rearrange option c to move the $x$ term to the same side as the $y$ term.
$-3x + y = 6$

STEP 10

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-3x + y = 6$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

STEP 11

Examine option d: $-8x+3y=12$
Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the equation $-8x+3y=12$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form according to the traditional definition, but some texts allow the coefficient of $x$ to be negative as long as it is an integer.

STEP 12

Determine which equation meets the standard form criteria.
Option a is not in standard form because the constant term is negative. Option b is not in standard form because the coefficient of $x$ is negative. Option c is not in standard form because the coefficient of $x$ is negative. Option d could be considered in standard form if the definition allows for a negative coefficient of $x$ .

STEP 13

If the definition of standard form allows for a negative coefficient of $x$ , then option d is the correct answer. Otherwise, none of the options are in perfect standard form according to the strictest definition.
The equation in standard form is option d: $-8x+3y=12$ .

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Determine which equation is in standard form: a. $x+3=-5y$ , b. $5-y=x$ , c. $y=3x+6$ , d. $-8x+3y=12$ .

STEP 1

Assumptions
1. Standard form for a linear equation is $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is non-negative.
2. We are given four options and we need to identify which one is in standard form.

STEP 2

Examine option a: $x+3=-5y$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 3

Rearrange option a to move the $y$ term to the same side as the $x$ term.
$x + 5y = -3$

STEP 4

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $x + 5y = -3$ , all coefficients are integers and the coefficient of $x$ is positive (1). However, the equation is not in standard form because the constant term is negative.

STEP 5

Examine option b: $5-y=x$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 6

Rearrange option b to move the $x$ term to the same side as the $y$ term.
$-x - y = -5$

STEP 7

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-x - y = -5$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

STEP 8

Examine option c: $y=3x+6$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

STEP 9

Rearrange option c to move the $x$ term to the same side as the $y$ term.
$-3x + y = 6$

STEP 10

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-3x + y = 6$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

STEP 11

STEP 12

STEP 13

If the definition of standard form allows for a negative coefficient of $x$ , then option d is the correct answer. Otherwise, none of the options are in perfect standard form according to the strictest definition.
The equation in standard form is option d: $-8x+3y=12$ .

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Determine which equation is in standard form: a. x+3=−5yx+3=-5yx+3=−5y, b. 5−y=x5-y=x5−y=x, c. y=3x+6y=3x+6y=3x+6, d. −8x+3y=12-8x+3y=12−8x+3y=12.

STEP 1

Assumptions1. Standard form for a linear equation is Ax+By=CAx + By = CAx+By=C, where AAA, BBB, and CCC are integers, and AAA is non-negative.2. We are given four options and we need to identify which one is in standard form.

STEP 2

Examine option a: x+3=−5yx+3=-5yx+3=−5yTo put this equation into standard form, we need to rearrange it so that xxx and yyy terms are on the same side and the constant is on the other side.

STEP 3

Rearrange option a to move the yyy term to the same side as the xxx term.x+5y=−3x + 5y = -3x+5y=−3

STEP 4

STEP 5

Examine option b: 5−y=x5-y=x5−y=xTo put this equation into standard form, we need to rearrange it so that xxx and yyy terms are on the same side and the constant is on the other side.

STEP 6

Rearrange option b to move the xxx term to the same side as the yyy term.−x−y=−5-x - y = -5−x−y=−5

STEP 7

Check if the coefficients are integers and if the coefficient of xxx is non-negative.In the rearranged equation −x−y=−5-x - y = -5−x−y=−5, all coefficients are integers but the coefficient of xxx is negative. This is not in standard form.

STEP 8

Examine option c: y=3x+6y=3x+6y=3x+6To put this equation into standard form, we need to rearrange it so that xxx and yyy terms are on the same side and the constant is on the other side.

STEP 9

Rearrange option c to move the xxx term to the same side as the yyy term.−3x+y=6-3x + y = 6−3x+y=6

STEP 10

Check if the coefficients are integers and if the coefficient of xxx is non-negative.In the rearranged equation −3x+y=6-3x + y = 6−3x+y=6, all coefficients are integers but the coefficient of xxx is negative. This is not in standard form.

STEP 11

STEP 12

STEP 13

If the definition of standard form allows for a negative coefficient of xxx, then option d is the correct answer. Otherwise, none of the options are in perfect standard form according to the strictest definition.The equation in standard form is option d: −8x+3y=12-8x+3y=12−8x+3y=12.

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Determine which equation is in standard form: a. $x+3=-5y$ , b. $5-y=x$ , c. $y=3x+6$ , d. $-8x+3y=12$ .

Assumptions
1. Standard form for a linear equation is $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is non-negative.
2. We are given four options and we need to identify which one is in standard form.

Examine option a: $x+3=-5y$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

Rearrange option a to move the $y$ term to the same side as the $x$ term.
$x + 5y = -3$

Examine option b: $5-y=x$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

Rearrange option b to move the $x$ term to the same side as the $y$ term.
$-x - y = -5$

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-x - y = -5$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

Examine option c: $y=3x+6$
To put this equation into standard form, we need to rearrange it so that $x$ and $y$ terms are on the same side and the constant is on the other side.

Rearrange option c to move the $x$ term to the same side as the $y$ term.
$-3x + y = 6$

Check if the coefficients are integers and if the coefficient of $x$ is non-negative.
In the rearranged equation $-3x + y = 6$ , all coefficients are integers but the coefficient of $x$ is negative. This is not in standard form.

If the definition of standard form allows for a negative coefficient of $x$ , then option d is the correct answer. Otherwise, none of the options are in perfect standard form according to the strictest definition.
The equation in standard form is option d: $-8x+3y=12$ .