The values of x and y from the equations x - y = 6 and x/3 + y/2 = 12 are:
x = 18, y = 12
Solving System of Linear Equations
We are given a system of two linear equations with two variables, x and y:
- \(x - y = 6\)
- \(x/3 + y/2 = 12\)
Our goal is to find the values of x and y that satisfy both equations simultaneously. We can use methods like substitution or elimination to solve this system.
Method 1: Substitution
Let's use the substitution method. From Equation 1, we can express x in terms of y:
\(x = y + 6\)
Now, substitute this expression for x into Equation 2:
\(\frac{(y + 6)}{3} + \frac{y}{2} = 12\)
To eliminate the denominators, find the least common multiple (LCM) of 3 and 2, which is 6. Multiply the entire equation by 6:
\(6 \times \left(\frac{y + 6}{3}\right) + 6 \times \left(\frac{y}{2}\right) = 6 \times 12\)
\(2(y + 6) + 3y = 72\)
Distribute the 2:
\(2y + 12 + 3y = 72\)
Combine like terms:
\(5y + 12 = 72\)
Subtract 12 from both sides:
\(5y = 72 - 12\)
\(5y = 60\)
Divide by 5 to find the value of y:
\(y = \frac{60}{5}\)
\(y = 12\)
Now that we have the value of y, substitute it back into the expression for x (\(x = y + 6\)):
\(x = 12 + 6\)
\(x = 18\)
Checking the Solution
Let's verify if \(x = 18\) and \(y = 12\) satisfy both original equations.
Equation 1: \(x - y = 6\)
\(18 - 12 = 6\)
\(6 = 6\) (True)
Equation 2: \(x/3 + y/2 = 12\)
\(\frac{18}{3} + \frac{12}{2} = 12\)
\(6 + 6 = 12\)
\(12 = 12\) (True)
Since the values \(x = 18\) and \(y = 12\) satisfy both equations, this is the correct solution for the system of linear equations.
| Equation | Substitute \(x=18, y=12\) | Result |
|---|---|---|
| \(x - y = 6\) | \(18 - 12\) | \(6 = 6\) (Holds true) |
| \(x/3 + y/2 = 12\) | \(18/3 + 12/2 = 6 + 6\) | \(12 = 12\) (Holds true) |
Conclusion
The values of x and y from the given equations \(x - y = 6\) and \(x/3 + y/2 = 12\) are \(x = 18\) and \(y = 12\).
Revision Table: System of Linear Equations
| Concept | Description | Methods to Solve |
|---|---|---|
| System of Linear Equations | A set of two or more linear equations involving the same variables. | Substitution, Elimination, Graphical Method |
| Solution to a System | The values of the variables that satisfy ALL equations in the system simultaneously. | Intersection point on a graph (for 2 variables) |
| Substitution Method | Solve one equation for one variable, then substitute that expression into the other equation. | Useful when one equation is easily solved for a variable. |
| Elimination Method | Multiply equations by constants so that the coefficients of one variable are opposites, then add the equations together to eliminate that variable. | Useful when coefficients are easy to make opposites. |
Additional Information: Linear Equations
A linear equation in two variables (like x and y) is an equation that can be written in the form \(Ax + By = C\), where A, B, and C are constants, and A and B are not both zero. The graph of a linear equation in two variables is a straight line.
When solving a system of two linear equations, there are three possible outcomes for the number of solutions:
- One unique solution: The lines intersect at exactly one point. (This is the case in our problem).
- No solution: The lines are parallel and distinct. They never intersect.
- Infinitely many solutions: The lines are the same line (coincident). Every point on the line is a solution.
Understanding how to solve systems of linear equations is fundamental in algebra and has applications in various fields, including science, engineering, economics, and computer science, to model and solve problems involving multiple related quantities.
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