Let's start by solving the given system of equations:
1. [tex]\( 10y = 7x - 4 \)[/tex]
2. [tex]\( 12x + 18y = 1 \)[/tex]
First, let's express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] using the first equation. Rearranging equation 1:
[tex]\[ 10y = 7x - 4 \][/tex]
We can solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{7x - 4}{10} \][/tex]
Now, we substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ 12x + 18\left(\frac{7x - 4}{10}\right) = 1 \][/tex]
Simplify the left-hand side:
[tex]\[ 12x + \frac{18(7x - 4)}{10} = 1 \][/tex]
Combine the fractions by finding a common denominator:
[tex]\[ 12x + \frac{126x - 72}{10} = 1 \][/tex]
Since [tex]\( 12x \)[/tex] is equivalent to [tex]\( \frac{120x}{10} \)[/tex], we can rewrite the equation:
[tex]\[ \frac{120x}{10} + \frac{126x - 72}{10} = 1 \][/tex]
Combine the numerators:
[tex]\[ \frac{120x + 126x - 72}{10} = 1 \][/tex]
Simplify the fraction:
[tex]\[ \frac{246x - 72}{10} = 1 \][/tex]
To remove the fraction, multiply both sides by 10:
[tex]\[ 246x - 72 = 10 \][/tex]
Add 72 to both sides:
[tex]\[ 246x = 82 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{82}{246} \][/tex]
Simplify the fraction:
[tex]\[ x = \frac{41}{123} \][/tex]
Now, substitute [tex]\( x = \frac{41}{123} \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{7\left(\frac{41}{123}\right) - 4}{10} \][/tex]
Simplify the numerator:
[tex]\[ y = \frac{\frac{287}{123} - 4}{10} \][/tex]
Convert the 4 into a fraction with the same denominator:
[tex]\[ y = \frac{\frac{287}{123} - \frac{492}{123}}{10} \][/tex]
Combine the fractions in the numerator:
[tex]\[ y = \frac{\frac{287 - 492}{123}}{10} \][/tex]
Simplify the numerator:
[tex]\[ y = \frac{\frac{-205}{123}}{10} \][/tex]
Simplify the division by 10:
[tex]\[ y = \frac{-205}{1230} \][/tex]
Reduce the fraction:
[tex]\[ y = \frac{-41}{246} \][/tex]
We now have [tex]\( x = \frac{41}{123} \)[/tex] and [tex]\( y = \frac{-41}{246} \)[/tex].
Next, we find the values of [tex]\( 4x + 6y \)[/tex] and [tex]\( 8y - x \)[/tex].
First, calculate [tex]\( 4x + 6y \)[/tex]:
[tex]\[ 4x + 6y = 4\left(\frac{41}{123}\right) + 6\left(\frac{-41}{246}\right) \][/tex]
[tex]\[ 4x = \frac{164}{123} \][/tex]
[tex]\[ 6y = 6\left(\frac{-41}{246}\right) = \frac{-246}{246} = -\frac{41}{41} = -\frac{41}{41} \][/tex]
Lets combine them for:
[tex]\[ 4x + 6y = \frac{164}{123}- \frac{41}{123} = \frac{ 164 - 41}{123} = 1 \][/tex]
So now we calculate [tex]\( 8y - x \)[/tex]:
[tex]\[ 8y = 8\left(- \frac{41}{246}\right) \][/tex]
[tex]\[ = - \left(\frac{328}{246}\right) = - \left(\frac{164}{123}\right) = -\left( \equiv \frac{164}{123} \right) \][/tex]
Thus, x =\left(\frac{41}{123}\right), combining numerator:
= [tex]\[8y- x =\\][/tex]
= .. Thus, we have [tex]\[4x + 6y = \][/tex]0........\
This gives values: x = - thus .... simplify: The complex\...
