Get the information you need quickly and easily with IDNLearn.com. Discover in-depth answers to your questions from our community of experienced professionals.
Sagot :
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\...
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\...
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
