2 A + B = 7 ... 2 B = 14 - 2 A
-A / B = 2 / 7 ... -7 A = 2 B
-7 A = 14 - 2 A
solve for A , substitute back to find B
Find B - A if the graph of Ax + By = 7 passes through (2,1) and is parallel to the graph of 2x - 7y = 3.
6 answers
For this I got -21/5 but it's wrong???? If this is wrong where did I go wrong?
sorry ... my mistake
2 B = 14 - 4 A ... not 2 A
-7 A = 14 - 4 A ... A = -14 / 3
B = 7 - 2 A = 49 / 3
B - A = 21
2 B = 14 - 4 A ... not 2 A
-7 A = 14 - 4 A ... A = -14 / 3
B = 7 - 2 A = 49 / 3
B - A = 21
2 x - 7 y = 3
Add 7 y to both sides
2 x - 7 y + 7 y = 3 + 7 y
2 x = 3 + 7 y
Subtract 3 to both sides
2 x - 3 = 3 + 7 y - 3
2 x - 3 = 7 y
7 y = 2 x - 3
Divide both sides by 7
y = ( 2 / 7 ) x - 3 / 7
Slope = 2 / 7
You also write this function like:
y = ( 1 / 7 ) ( 2x - 3 )
Now:
A x + B y = 7
Subtract A x to both sides
A x + B y - A x = 7 - A x
B y = 7 - A x
B y = - A x + 7
Divide both sides by B
y = - ( A / B ) x + 7
Parallel lines means it has the same slope.
- A / B = 2 / 7
Multiply both sides by B
- A = ( 2 / 7 ) B
Multiply both sides by - 1
A = ( - 2 / 7 ) B
A x + B y = 7
( - 2 / 7 ) B ∙ x + B y = 7
Passes through (2,1) mean:
x = 2 , y = 1
( - 2 / 7 ) B ∙ x + B ∙ y = 7
( - 2 / 7 ) B ∙ 2 + B ∙ 1 = 7
( - 4 / 7 ) B + B = 7
( - 4 / 7 ) B + ( 7 / 7 ) B = 7
( 3 / 7 ) B = 7
Multiply both sides by 7
3 B = 7 ∙ 7
3 B = 49
Divide both sides by 3
B = 49 / 3
A = ( - 2 / 7 ) B
A = ( - 2 / 7 ) ∙ ( 49 / 3 )
A = ( - 2 ∙ 49 ) / ( 7 ∙ 3 )
A = - 98 / 21
A = - 14 ∙ 7 / 3 ∙ 7
A = - 14 / 3
A = - 14 / 3
B = 49 / 3
B - A = B = 49 / 3 - ( - 14 / 3 ) = 49 / 3 + 14 / 3 = B = 63 / 3 = 21
Also:
A x + B y = 7
( - 14 / 3 ) x + ( 49 / 3 ) y = 7
( - 7 ∙ 2 / 3 ) x + ( 7 ∙ 7 / 3 ) y = 7
( 7 / 3 ) ∙ [ ( - 2 x ) + 7 y ] = 7
( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7
This mean:
A x + B y = 7
is
( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7
If you want go on:
wolframalpha.c o m
When page be open type:
plot y = ( 1 / 7 ) ( 2 x - 3 ) , ( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7 , x = 2 , y = 1
you will see graph
Add 7 y to both sides
2 x - 7 y + 7 y = 3 + 7 y
2 x = 3 + 7 y
Subtract 3 to both sides
2 x - 3 = 3 + 7 y - 3
2 x - 3 = 7 y
7 y = 2 x - 3
Divide both sides by 7
y = ( 2 / 7 ) x - 3 / 7
Slope = 2 / 7
You also write this function like:
y = ( 1 / 7 ) ( 2x - 3 )
Now:
A x + B y = 7
Subtract A x to both sides
A x + B y - A x = 7 - A x
B y = 7 - A x
B y = - A x + 7
Divide both sides by B
y = - ( A / B ) x + 7
Parallel lines means it has the same slope.
- A / B = 2 / 7
Multiply both sides by B
- A = ( 2 / 7 ) B
Multiply both sides by - 1
A = ( - 2 / 7 ) B
A x + B y = 7
( - 2 / 7 ) B ∙ x + B y = 7
Passes through (2,1) mean:
x = 2 , y = 1
( - 2 / 7 ) B ∙ x + B ∙ y = 7
( - 2 / 7 ) B ∙ 2 + B ∙ 1 = 7
( - 4 / 7 ) B + B = 7
( - 4 / 7 ) B + ( 7 / 7 ) B = 7
( 3 / 7 ) B = 7
Multiply both sides by 7
3 B = 7 ∙ 7
3 B = 49
Divide both sides by 3
B = 49 / 3
A = ( - 2 / 7 ) B
A = ( - 2 / 7 ) ∙ ( 49 / 3 )
A = ( - 2 ∙ 49 ) / ( 7 ∙ 3 )
A = - 98 / 21
A = - 14 ∙ 7 / 3 ∙ 7
A = - 14 / 3
A = - 14 / 3
B = 49 / 3
B - A = B = 49 / 3 - ( - 14 / 3 ) = 49 / 3 + 14 / 3 = B = 63 / 3 = 21
Also:
A x + B y = 7
( - 14 / 3 ) x + ( 49 / 3 ) y = 7
( - 7 ∙ 2 / 3 ) x + ( 7 ∙ 7 / 3 ) y = 7
( 7 / 3 ) ∙ [ ( - 2 x ) + 7 y ] = 7
( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7
This mean:
A x + B y = 7
is
( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7
If you want go on:
wolframalpha.c o m
When page be open type:
plot y = ( 1 / 7 ) ( 2 x - 3 ) , ( - 7 / 3 ) ∙ ( 2 x - 7 y ) = 7 , x = 2 , y = 1
you will see graph
Solution:
Since the graph of $Ax + By = 7$ passes through $(x,y)=(2,1)$, we have
\[2A + B = 7.\]
Two lines in the Cartesian plane are parallel if their slopes are equal. We find the slope of each line by solving each equation for $y$, thereby putting each equation in slope-intercept form. This gives us
\begin{align*}
y &= -\frac{A}{B}x + \frac7B,\\
y&= \frac27x - \frac37.
\end{align*}
The slopes then are $-\frac{A}{B}$ and $\frac27$, respectively, so we must have $-\frac{A}{B} = \frac27$. Therefore, we have the following system of equations:
\begin{align*}
2A+B&=7,\\
-\frac{A}{B}&=\frac27.\end{align*}
Solving the first equation for $B$ gives $B = 7-2A$. Substituting this into the second equation gives
\[-\frac{A}{7-2A} = \frac27.\]
Multiplying both sides by 7(7-2A) gives -7A = 2(7-2A). Expanding the right-hand side gives $-7A = 14 -4A$, so $-3A = 14$ and $A = -\frac{14}{3}$. Therefore, $B = 7-2A = 7 +\frac{28}{3} = \frac{49}{3}$ and B-A = \frac{49}{3}-\left(-\frac{14}{3}\right) = \frac{63}{3} =21
the \frac means fraction.
Since the graph of $Ax + By = 7$ passes through $(x,y)=(2,1)$, we have
\[2A + B = 7.\]
Two lines in the Cartesian plane are parallel if their slopes are equal. We find the slope of each line by solving each equation for $y$, thereby putting each equation in slope-intercept form. This gives us
\begin{align*}
y &= -\frac{A}{B}x + \frac7B,\\
y&= \frac27x - \frac37.
\end{align*}
The slopes then are $-\frac{A}{B}$ and $\frac27$, respectively, so we must have $-\frac{A}{B} = \frac27$. Therefore, we have the following system of equations:
\begin{align*}
2A+B&=7,\\
-\frac{A}{B}&=\frac27.\end{align*}
Solving the first equation for $B$ gives $B = 7-2A$. Substituting this into the second equation gives
\[-\frac{A}{7-2A} = \frac27.\]
Multiplying both sides by 7(7-2A) gives -7A = 2(7-2A). Expanding the right-hand side gives $-7A = 14 -4A$, so $-3A = 14$ and $A = -\frac{14}{3}$. Therefore, $B = 7-2A = 7 +\frac{28}{3} = \frac{49}{3}$ and B-A = \frac{49}{3}-\left(-\frac{14}{3}\right) = \frac{63}{3} =21
the \frac means fraction.
Also ignore the $ signs.
1 answer