Solving Systems of Linear Equations

We are given a system of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously.

The given equations are:

1. $4x + 5y = 14$
2. $x - 5y = 16$

We can solve this system using methods like substitution or elimination. The elimination method seems straightforward here because the coefficients of 'y' in both equations are opposites (+5y and -5y). By adding the two equations, the 'y' term will be eliminated.

Step-by-Step Solution to Find x and y

Step 1: Eliminate one variable.

Let's add Equation (1) and Equation (2):

$(4x + 5y) + (x - 5y) = 14 + 16$

$4x + x + 5y - 5y = 30$

$5x + 0y = 30$

$5x = 30$

Step 2: Solve for the remaining variable (x).

Divide both sides by 5:

$\frac{5x}{5} = \frac{30}{5}$

$x = 6$

So, the value of x is 6.

Step 3: Substitute the value of x into one of the original equations to find y.

Let's use Equation (2): $x - 5y = 16$

Substitute $x = 6$ into this equation:

$6 - 5y = 16$

Step 4: Solve for y.

Subtract 6 from both sides:

$-5y = 16 - 6$

$-5y = 10$

Divide both sides by -5:

$\frac{-5y}{-5} = \frac{10}{-5}$

$y = -2$

So, the value of y is -2.

The values of x and y that satisfy the system of equations are $x = 6$ and $y = -2$.

Verifying the Solution

Let's check if these values satisfy both original equations:

Equation 1: $4x + 5y = 14$

$4(6) + 5(-2) = 24 - 10 = 14$. This is correct.

Equation 2: $x - 5y = 16$

$6 - 5(-2) = 6 - (-10) = 6 + 10 = 16$. This is also correct.

The solution $(x, y) = (6, -2)$ is correct.

Comparing with Options

Let's look at the given options:

• A. 10 and –6/5 (x=10, y=-6/5)
• B. 6 and 2 (x=6, y=2)
• C. 10 and 6/5 (x=10, y=6/5)
• D. 6 and – 2 (x=6, y=-2)

Our calculated values are $x=6$ and $y=-2$, which match Option D.

Therefore, the correct values for x and y are 6 and -2.

Revision Table: Key Concepts for Solving Linear Equations

Additional Information: Systems of Equations

A system of linear equations can have:

• Exactly one solution: The lines intersect at a single point (as in this problem). This is a consistent and independent system.
• No solution: The lines are parallel and never intersect. This is an inconsistent system.
• Infinitely many solutions: The equations represent the same line. This is a consistent and dependent system.

Understanding these possibilities helps in interpreting the results when solving systems of equations.