Solved on Feb 19, 2024

Solve the system of linear equations with constants $b$ and $c$ . If $b = c - \frac{1}{2}$ , determine which statement about $x$ and $y$ is true.

STEP 1

Assumptions
1. We have two equations: $3x + b = 5x - 7$ and $3y + c = 5y - 7$ .
2. $b$ and $c$ are constants.
3. $b$ is $c$ minus $\frac{1}{2}$ , i.e., $b = c - \frac{1}{2}$ .
4. We need to determine the relationship between $x$ and $y$ .

STEP 2

First, we will solve the equation $3x + b = 5x - 7$ for $x$ .
$3x + b = 5x - 7$

STEP 3

Subtract $3x$ from both sides of the equation to move the $x$ terms to one side.
$b = 2x - 7$

STEP 4

Now, solve the equation $3y + c = 5y - 7$ for $y$ .
$3y + c = 5y - 7$

STEP 5

Subtract $3y$ from both sides of the equation to move the $y$ terms to one side.
$c = 2y - 7$

STEP 6

Using the relationship between $b$ and $c$ , we can set up the following equation:
$b = c - \frac{1}{2}$

STEP 7

Substitute $b$ and $c$ from the equations found in STEP_3 and STEP_5 into the equation from STEP_6.
$2x - 7 = (2y - 7) - \frac{1}{2}$

STEP 8

Simplify the right side of the equation.
$2x - 7 = 2y - 7 - \frac{1}{2}$

STEP 9

Add $7$ to both sides of the equation to eliminate the constant term.
$2x = 2y - \frac{1}{2}$

STEP 10

Divide both sides of the equation by $2$ to solve for $x$ in terms of $y$ .
$x = y - \frac{1}{4}$
Therefore, the correct answer is: A) $x$ is $y$ minus $\frac{1}{4}$ .

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Solve the system of linear equations with constants $b$ and $c$ . If $b = c - \frac{1}{2}$ , determine which statement about $x$ and $y$ is true.

STEP 1

Assumptions
1. We have two equations: $3x + b = 5x - 7$ and $3y + c = 5y - 7$ .
2. $b$ and $c$ are constants.
3. $b$ is $c$ minus $\frac{1}{2}$ , i.e., $b = c - \frac{1}{2}$ .
4. We need to determine the relationship between $x$ and $y$ .

STEP 2

First, we will solve the equation $3x + b = 5x - 7$ for $x$ .
$3x + b = 5x - 7$

STEP 3

Subtract $3x$ from both sides of the equation to move the $x$ terms to one side.
$b = 2x - 7$

STEP 4

Now, solve the equation $3y + c = 5y - 7$ for $y$ .
$3y + c = 5y - 7$

STEP 5

Subtract $3y$ from both sides of the equation to move the $y$ terms to one side.
$c = 2y - 7$

STEP 6

Using the relationship between $b$ and $c$ , we can set up the following equation:
$b = c - \frac{1}{2}$

STEP 7

Substitute $b$ and $c$ from the equations found in STEP_3 and STEP_5 into the equation from STEP_6.
$2x - 7 = (2y - 7) - \frac{1}{2}$

STEP 8

Simplify the right side of the equation.
$2x - 7 = 2y - 7 - \frac{1}{2}$

STEP 9

Add $7$ to both sides of the equation to eliminate the constant term.
$2x = 2y - \frac{1}{2}$

STEP 10

Divide both sides of the equation by $2$ to solve for $x$ in terms of $y$ .
$x = y - \frac{1}{4}$
Therefore, the correct answer is: A) $x$ is $y$ minus $\frac{1}{4}$ .

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Solve the system of linear equations with constants bbb and ccc. If b=c−12b = c - \frac{1}{2}b=c−21​, determine which statement about xxx and yyy is true.

STEP 1

Assumptions1. We have two equations: 3x+b=5x−73x + b = 5x - 73x+b=5x−7 and 3y+c=5y−73y + c = 5y - 73y+c=5y−7.2. bbb and ccc are constants.3. bbb is ccc minus 12\frac{1}{2}21​, i.e., b=c−12b = c - \frac{1}{2}b=c−21​.4. We need to determine the relationship between xxx and yyy.

STEP 2

First, we will solve the equation 3x+b=5x−73x + b = 5x - 73x+b=5x−7 for xxx.3x+b=5x−73x + b = 5x - 73x+b=5x−7

STEP 3

Subtract 3x3x3x from both sides of the equation to move the xxx terms to one side.b=2x−7b = 2x - 7b=2x−7

STEP 4

Now, solve the equation 3y+c=5y−73y + c = 5y - 73y+c=5y−7 for yyy.3y+c=5y−73y + c = 5y - 73y+c=5y−7

STEP 5

Subtract 3y3y3y from both sides of the equation to move the yyy terms to one side.c=2y−7c = 2y - 7c=2y−7

STEP 6

Using the relationship between bbb and ccc, we can set up the following equation:b=c−12b = c - \frac{1}{2}b=c−21​

STEP 7

Substitute bbb and ccc from the equations found in STEP_3 and STEP_5 into the equation from STEP_6.2x−7=(2y−7)−122x - 7 = (2y - 7) - \frac{1}{2}2x−7=(2y−7)−21​

STEP 8

Simplify the right side of the equation.2x−7=2y−7−122x - 7 = 2y - 7 - \frac{1}{2}2x−7=2y−7−21​

STEP 9

Add 777 to both sides of the equation to eliminate the constant term.2x=2y−122x = 2y - \frac{1}{2}2x=2y−21​

STEP 10

Divide both sides of the equation by 222 to solve for xxx in terms of yyy.x=y−14x = y - \frac{1}{4}x=y−41​Therefore, the correct answer is: A) xxx is yyy minus 14\frac{1}{4}41​.

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Solve the system of linear equations with constants $b$ and $c$ . If $b = c - \frac{1}{2}$ , determine which statement about $x$ and $y$ is true.

Assumptions
1. We have two equations: $3x + b = 5x - 7$ and $3y + c = 5y - 7$ .
2. $b$ and $c$ are constants.
3. $b$ is $c$ minus $\frac{1}{2}$ , i.e., $b = c - \frac{1}{2}$ .
4. We need to determine the relationship between $x$ and $y$ .

First, we will solve the equation $3x + b = 5x - 7$ for $x$ .
$3x + b = 5x - 7$

Subtract $3x$ from both sides of the equation to move the $x$ terms to one side.
$b = 2x - 7$

Now, solve the equation $3y + c = 5y - 7$ for $y$ .
$3y + c = 5y - 7$

Subtract $3y$ from both sides of the equation to move the $y$ terms to one side.
$c = 2y - 7$

Using the relationship between $b$ and $c$ , we can set up the following equation:
$b = c - \frac{1}{2}$

Substitute $b$ and $c$ from the equations found in STEP_3 and STEP_5 into the equation from STEP_6.
$2x - 7 = (2y - 7) - \frac{1}{2}$

Simplify the right side of the equation.
$2x - 7 = 2y - 7 - \frac{1}{2}$

Add $7$ to both sides of the equation to eliminate the constant term.
$2x = 2y - \frac{1}{2}$

Divide both sides of the equation by $2$ to solve for $x$ in terms of $y$ .
$x = y - \frac{1}{4}$
Therefore, the correct answer is: A) $x$ is $y$ minus $\frac{1}{4}$ .