What is the solution of the following equations ? 2x + 3y = 12 and 3x − 2y = 5
x = 3, y = 2
Solving Systems of Linear Equations: 2x + 3y = 12 and 3x - 2y = 5
Let's solve the given system of linear equations. We have two equations with two variables, x and y:
Equation 1: ${2x + 3y = 12}$
Equation 2: ${3x - 2y = 5}$
We can use the elimination method to find the values of x and y that satisfy both equations simultaneously.
Step-by-Step Solution using Elimination Method
The goal of the elimination method is to eliminate one variable by making its coefficients in both equations equal in magnitude but opposite in sign, and then adding the equations.
Step 1: Make the coefficients of one variable the same magnitude.
Let's choose to eliminate 'y'. The coefficients of 'y' are +3 and -2. The least common multiple of 3 and 2 is 6. We can make the coefficients +6y and -6y.
- Multiply Equation 1 by 2: ${2 \times (2x + 3y) = 2 \times 12}$
- This gives: ${4x + 6y = 24}$ (Let's call this Equation 3)
- Multiply Equation 2 by 3: ${3 \times (3x - 2y) = 3 \times 5}$
- This gives: ${9x - 6y = 15}$ (Let's call this Equation 4)
Step 2: Add the modified equations.
Now, add Equation 3 and Equation 4. Notice that the '+6y' and '-6y' terms will cancel out.
${(4x + 6y) + (9x - 6y) = 24 + 15}$
${4x + 9x + 6y - 6y = 39}$
${13x = 39}$
Step 3: Solve for the remaining variable (x).
Divide both sides of the equation by 13:
${13x / 13 = 39 / 13}$
${x = 3}$
Step 4: Substitute the value of x into one of the original equations to solve for y.
Let's substitute ${x = 3}$ into Equation 1 (${2x + 3y = 12}$):
${2 \times (3) + 3y = 12}$
${6 + 3y = 12}$
Subtract 6 from both sides:
${3y = 12 - 6}$
${3y = 6}$
Divide by 3:
${3y / 3 = 6 / 3}$
${y = 2}$
Solution
The solution to the system of equations is ${x = 3}$ and ${y = 2}$.
Verification
Let's check if these values satisfy both original equations:
- Equation 1: ${2x + 3y = 12}$
- Substitute ${x = 3}$ and ${y = 2}$: ${2(3) + 3(2) = 6 + 6 = 12}$
- ${12 = 12}$ (True)
- Equation 2: ${3x - 2y = 5}$
- Substitute ${x = 3}$ and ${y = 2}$: ${3(3) - 2(2) = 9 - 4 = 5}$
- ${5 = 5}$ (True)
Since the values ${x = 3}$ and ${y = 2}$ satisfy both equations, this is the correct solution.
| Equation | Substitute ${x=3, y=2}$ | Result |
|---|---|---|
| ${2x + 3y = 12}$ | ${2(3) + 3(2) = 6 + 6}$ | ${12}$ (Matches RHS) |
| ${3x - 2y = 5}$ | ${3(3) - 2(2) = 9 - 4}$ | ${5}$ (Matches RHS) |
Revision Table: Solving Linear Equations
| Method | Description | When to Use |
|---|---|---|
| Elimination | Multiply equations to match coefficients of one variable, then add or subtract to eliminate it. | Useful when coefficients are easy to make equal or opposite, or for more complex systems. |
| Substitution | Solve one equation for one variable, then substitute that expression into the other equation. | Useful when one equation is already solved for a variable, or has a coefficient of 1 or -1. |
| Graphical | Graph both equations. The intersection point is the solution. | Good for visualizing, but can be inaccurate for non-integer solutions. |
Additional Information on Systems of Equations
A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system is the set of values for the variables that satisfies all equations in the system simultaneously.
Based on the number of solutions, a system of linear equations can be classified as:
- Consistent System: Has at least one solution.
- Independent: Has exactly one solution (like the example solved here). The lines intersect at one point.
- Dependent: Has infinitely many solutions. The equations represent the same line.
- Inconsistent System: Has no solution. The lines are parallel and distinct.
Solving systems of equations is a fundamental concept in algebra and has applications in various fields, including science, engineering, economics, and computer science.
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