Get expert advice and community support on IDNLearn.com. Join our community to access reliable and comprehensive responses to your questions from experienced professionals.
Sagot :
To solve for the quotient [tex]\(\frac{x}{y}\)[/tex] given the system of equations:
1. [tex]\[-2x + 2 = 9y - 4x\][/tex]
2. [tex]\[2x + y = -6\][/tex]
Let's first work on simplifying and solving these equations.
### Step 1: Simplify the First Equation
Starting with the first equation:
[tex]\[ -2x + 2 = 9y - 4x \][/tex]
Add [tex]\(4x\)[/tex] to both sides to get:
[tex]\[ 2x + 2 = 9y \][/tex]
Subtract 2 from both sides to isolate terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ 2x = 9y - 2 \][/tex]
We can rewrite it for simplicity:
[tex]\[ 2x - 9y = -2 \quad \text{(Equation 3)} \][/tex]
### Step 2: Simplify the Second Equation
Next, consider the second equation:
[tex]\[ 2x + y = -6 \quad \text{(Equation 2)} \][/tex]
### Step 3: Solve the System of Equations
We have the two equations:
1. [tex]\(2x - 9y = -2\)[/tex]
2. [tex]\(2x + y = -6\)[/tex]
Let's solve the system using substitution or elimination. We'll use the substitution method.
#### Isolate [tex]\(y\)[/tex] from Equation 2:
[tex]\[ 2x + y = -6 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -6 - 2x \][/tex]
#### Substitute [tex]\(y\)[/tex] in Equation 3:
[tex]\[ 2x - 9(-6 - 2x) = -2 \][/tex]
Simplify within the brackets:
[tex]\[ 2x + 54 + 18x = -2 \][/tex]
Combine like terms:
[tex]\[ 20x + 54 = -2 \][/tex]
Subtract 54 from both sides:
[tex]\[ 20x = -56 \][/tex]
Divide by 20:
[tex]\[ x = -\frac{56}{20} = -\frac{14}{5} \][/tex]
#### Substitute [tex]\(x\)[/tex] back into the isolated [tex]\(y\)[/tex] equation:
[tex]\[ y = -6 - 2\left(-\frac{14}{5}\right) \][/tex]
Simplify the expression:
[tex]\[ y = -6 + \frac{28}{5} \][/tex]
To combine the terms, convert [tex]\(-6\)[/tex] to a fraction with the same denominator:
[tex]\[ -6 = \frac{-30}{5} \][/tex]
Thus:
[tex]\[ y = \frac{-30}{5} + \frac{28}{5} = \frac{-30 + 28}{5} = \frac{-2}{5} \][/tex]
### Step 4: Calculate the Quotient [tex]\(\frac{x}{y}\)[/tex]
Now that we have the values:
[tex]\[ x = -\frac{14}{5}, \quad y = -\frac{2}{5} \][/tex]
The quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[ \frac{x}{y} = \frac{-\frac{14}{5}}{-\frac{2}{5}} = \frac{14}{5} \times \frac{5}{2} = \frac{14}{2} = 7 \][/tex]
### Conclusion
The value of the quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[ \boxed{7} \][/tex]
1. [tex]\[-2x + 2 = 9y - 4x\][/tex]
2. [tex]\[2x + y = -6\][/tex]
Let's first work on simplifying and solving these equations.
### Step 1: Simplify the First Equation
Starting with the first equation:
[tex]\[ -2x + 2 = 9y - 4x \][/tex]
Add [tex]\(4x\)[/tex] to both sides to get:
[tex]\[ 2x + 2 = 9y \][/tex]
Subtract 2 from both sides to isolate terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ 2x = 9y - 2 \][/tex]
We can rewrite it for simplicity:
[tex]\[ 2x - 9y = -2 \quad \text{(Equation 3)} \][/tex]
### Step 2: Simplify the Second Equation
Next, consider the second equation:
[tex]\[ 2x + y = -6 \quad \text{(Equation 2)} \][/tex]
### Step 3: Solve the System of Equations
We have the two equations:
1. [tex]\(2x - 9y = -2\)[/tex]
2. [tex]\(2x + y = -6\)[/tex]
Let's solve the system using substitution or elimination. We'll use the substitution method.
#### Isolate [tex]\(y\)[/tex] from Equation 2:
[tex]\[ 2x + y = -6 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -6 - 2x \][/tex]
#### Substitute [tex]\(y\)[/tex] in Equation 3:
[tex]\[ 2x - 9(-6 - 2x) = -2 \][/tex]
Simplify within the brackets:
[tex]\[ 2x + 54 + 18x = -2 \][/tex]
Combine like terms:
[tex]\[ 20x + 54 = -2 \][/tex]
Subtract 54 from both sides:
[tex]\[ 20x = -56 \][/tex]
Divide by 20:
[tex]\[ x = -\frac{56}{20} = -\frac{14}{5} \][/tex]
#### Substitute [tex]\(x\)[/tex] back into the isolated [tex]\(y\)[/tex] equation:
[tex]\[ y = -6 - 2\left(-\frac{14}{5}\right) \][/tex]
Simplify the expression:
[tex]\[ y = -6 + \frac{28}{5} \][/tex]
To combine the terms, convert [tex]\(-6\)[/tex] to a fraction with the same denominator:
[tex]\[ -6 = \frac{-30}{5} \][/tex]
Thus:
[tex]\[ y = \frac{-30}{5} + \frac{28}{5} = \frac{-30 + 28}{5} = \frac{-2}{5} \][/tex]
### Step 4: Calculate the Quotient [tex]\(\frac{x}{y}\)[/tex]
Now that we have the values:
[tex]\[ x = -\frac{14}{5}, \quad y = -\frac{2}{5} \][/tex]
The quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[ \frac{x}{y} = \frac{-\frac{14}{5}}{-\frac{2}{5}} = \frac{14}{5} \times \frac{5}{2} = \frac{14}{2} = 7 \][/tex]
### Conclusion
The value of the quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[ \boxed{7} \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
