Y 5x 7 3x 2y 12

Solving the System of Equations: y = 5x + 7 and 3x + 2y = 12

This article provides a comprehensive guide on how to solve the system of equations: y = 5x + 7 and 3x + 2y = 12. We'll explore multiple methods, delve into the underlying mathematical concepts, and offer practical tips for similar problems. Understanding systems of equations is fundamental in algebra and has widespread applications in various fields.

Understanding Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. In our case, we have two linear equations, meaning the highest power of the variables is 1. These equations represent lines on a coordinate plane, and the solution represents the point where these lines intersect.

The Equations:

• Equation 1: y = 5x + 7 (This is already in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept)
• Equation 2: 3x + 2y = 12 (This is in standard form: Ax + By = C)

Method 1: Substitution Method

This method involves solving one equation for one variable and substituting that expression into the other equation. Since Equation 1 is already solved for 'y', this is the easiest approach for this particular system.

Steps:

1. Substitute: Substitute the expression for 'y' from Equation 1 (y = 5x + 7) into Equation 2: 3x + 2(5x + 7) = 12

2. Simplify and Solve for x: Expand and simplify the equation: 3x + 10x + 14 = 12 13x = 12 - 14 13x = -2 x = -2/13

3. Substitute x to find y: Substitute the value of x (x = -2/13) back into either Equation 1 or Equation 2 to solve for y. Let's use Equation 1: y = 5(-2/13) + 7 y = -10/13 + 7 y = (-10 + 91)/13 y = 81/13

Solution: Therefore, the solution to the system of equations is x = -2/13 and y = 81/13. This represents the point of intersection of the two lines on the coordinate plane.

Method 2: Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations to eliminate one variable by adding or subtracting the equations.

Steps:

1. Prepare the Equations: Multiply Equation 1 by -2 to make the coefficients of 'y' opposites: -2(y) = -2(5x + 7) -2y = -10x - 14

2. Add the Equations: Add the modified Equation 1 to Equation 2: 3x + 2y = 12 -2y = -10x - 14

   3x - 2y + 2y = 12 -10x -14 3x = -10x - 2

3. Solve for x: Simplify and solve for x: 13x = -2 x = -2/13

4. Substitute x to find y: Substitute the value of x (x = -2/13) into either Equation 1 or Equation 2 to solve for y. (We'll use Equation 1 again for consistency): y = 5(-2/13) + 7 y = 81/13

Solution: Again, we arrive at the solution x = -2/13 and y = 81/13.

Method 3: Graphical Method

This method involves graphing both equations on a coordinate plane and identifying the point of intersection. While less precise than algebraic methods for fractional solutions, it provides a visual representation of the solution.

Steps:

1. Graph Equation 1: y = 5x + 7. This is a line with a y-intercept of 7 and a slope of 5 (meaning it rises 5 units for every 1 unit it moves to the right).

2. Graph Equation 2: 3x + 2y = 12. You can rewrite this in slope-intercept form (y = -3x/2 + 6) to make graphing easier. This line has a y-intercept of 6 and a slope of -3/2.

3. Find the Intersection Point: The point where the two lines intersect represents the solution to the system of equations. Because the solution involves fractions, accurately identifying the intersection point graphically can be challenging without sophisticated graphing tools.

Verifying the Solution

It's crucial to verify the solution by substituting the values of x and y back into both original equations.

Verification using Equation 1: y = 5x + 7

81/13 = 5(-2/13) + 7 81/13 = -10/13 + 91/13 81/13 = 81/13 (True)

Verification using Equation 2: 3x + 2y = 12

3(-2/13) + 2(81/13) = 12 -6/13 + 162/13 = 12 156/13 = 12 12 = 12 (True)

Since the solution satisfies both equations, our solution is correct.

Applications of Systems of Equations

Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications, including:

• Linear Programming: Optimizing resource allocation in businesses and industries.
• Engineering: Designing structures and analyzing forces.
• Economics: Modeling supply and demand, and predicting market equilibrium.
• Physics: Solving problems involving motion, forces, and energy.
• Computer Graphics: Creating realistic images and animations.

Understanding and mastering techniques to solve systems of equations is a valuable skill across various disciplines.

Tips for Solving Systems of Equations

• Choose the Best Method: The substitution method is usually easiest when one equation is already solved for a variable. The elimination method is efficient when coefficients are easily manipulated to create opposites. The graphical method provides a visual understanding but might lack precision for non-integer solutions.

• Check Your Work: Always substitute your solution back into the original equations to verify its accuracy.

• Practice Regularly: Solving systems of equations requires practice. Work through various examples to build your skills and confidence.

• Use Technology Wisely: Graphing calculators and software can assist with graphing and solving systems, especially those with complex solutions. However, it's important to understand the underlying mathematical principles to interpret the results correctly.

Conclusion

Solving the system of equations y = 5x + 7 and 3x + 2y = 12 demonstrates the fundamental principles of solving simultaneous equations. We explored three different methods—substitution, elimination, and graphical—highlighting their strengths and weaknesses. The solution, x = -2/13 and y = 81/13, was verified using both original equations. Understanding these methods and their applications is crucial for success in algebra and various other fields. Remember to practice consistently to solidify your understanding and problem-solving abilities. By mastering these techniques, you’ll be well-equipped to tackle more complex systems of equations in the future.