Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-6 y=7 \\ 5 x-2 y=5 \end{array}\right. $$

Short answer

x = 2/3, y = -5/6

Step-by-step solution

Label the Equations

Label the given equations for ease of reference. Let's call them: Equation 1: 3x - 6y = 7 Equation 2: 5x - 2y = 5

Multiply Equations for Elimination

To eliminate one variable, make the coefficients of either x or y the same in both equations. Multiply Equation 1 by 5 and Equation 2 by 3 to align the coefficients of x.Equation 1: 5(3x - 6y) = 5(7) 15x - 30y = 35Equation 2: 3(5x - 2y) = 3(5) 15x - 6y = 15

Subtract the Equations

Subtract the modified Equation 2 from the modified Equation 1 to eliminate x.(15x - 30y) - (15x - 6y) = 35 - 15This simplifies to:15x - 30y - 15x + 6y = 20Combine like terms:-24y = 20

Solve for y

Divide both sides of the equation by -24 to find y.y = 20 / -24 y = -5/6

Substitute y into One of the Original Equations

Use the value of y in one of the original equations to solve for x. Substitute y = -5/6 into Equation 1.3x - 6(-5/6) = 7Simplify:3x + 5 = 7Solve for x:3x = 2x = 2/3

Verify the Solution

Substitute x = 2/3 and y = -5/6 into both original equations to ensure they hold true.Checking Equation 1: 3(2/3) - 6(-5/6) = 7Simplifies to:2 + 5 = 77 = 7 (True)Checking Equation 2: 5(2/3) - 2(-5/6) = 5Simplifies to:10/3 + 5/3 = 5(10 + 5) / 3 = 515/3 = 55 = 5 (True)The solution satisfies both equations.

Key concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. It involves adjusting the equations to eliminate one variable, making it easier to solve for the other. Here's a simple overview of how the elimination method works:
First, you must manipulate the equations so that one of the variables cancels out. You achieve this by multiplying one or both equations by a constant, aligning the coefficients of one variable. In this exercise, we multiplied the first equation by 5 and the second by 3 to align the coefficients of x.
Next, you subtract one equation from the other. The variable with the aligned coefficients drops out, leaving you with a single-variable equation.
Finally, you solve the remaining equation for the single variable, then substitute this value back into one of the original equations to find the value of the second variable. The elimination method is efficient and often straightforward for many systems of linear equations.

Consistent System

A consistent system of equations has at least one solution that satisfies all equations in the system simultaneously. In this exercise, we checked the solutions by substituting back into the original equations to ensure they held true.
Systems of equations can either be consistent or inconsistent. If a system is consistent, it may have a unique solution or infinitely many solutions. If it is inconsistent, no solution exists that satisfies all equations at once.
This system turned out to be consistent because we found values of x and y that satisfied both equations. It's critical to verify your solution by plugging it back into the original equations to confirm the system's consistency.

Solving Linear Equations

Solving linear equations involves finding the values of the unknown variables that make the equation true. Here are simple steps to follow:

• Isolate the variable on one side of the equation.
• Simplify both sides of the equation if necessary.
• Perform inverse operations to solve for the variable.

In this exercise, we solved one of the equations using the elimination method to isolate and find the value of y. Then, we substituted this value into the other equation to find x. Always check your solution by substituting the values back into the original equations to verify your work.

Substitution Method

The substitution method is another technique for solving systems of linear equations. It involves solving one of the equations for one variable and substituting this expression into the other equation. While not used as the primary method in this exercise, it's good to understand how it works.
Here's how you apply the substitution method:

• Solve one equation for one variable in terms of the other variable.
• Substitute the expression from step 1 into the other equation.
• Solve the resulting single-variable equation.
• Substitute the solution back into the first equation to find the second variable.

Using the substitution method can be advantageous in cases where one equation is already easily solvable for one variable. Comparing it with elimination helps to build strong problem-solving skills.

Algebraic Solutions

Algebraic solutions are methods used to solve equations and systems of equations without graphing. They rely on algebraic manipulations and properties to find the unknown variable values.
The two primary algebraic methods for solving systems are elimination and substitution. Both methods involve:

• Using arithmetic operations to manipulate the equations.
• Isolating variables to find their values.
• Checking solutions by substituting them back into the original equations.

Mastering algebraic solutions is essential because it allows you to solve complex systems accurately and efficiently. Moreover, it forms the foundation for more advanced mathematics and applications.