Find the solution of the system of equations: 8x - 2y = -30 and -9x + 4y = 18.

Understand the Problem

The question is asking to solve a system of two linear equations in two variables, x and y. To solve it, we will use methods such as substitution or elimination to find the values of x and y that satisfy both equations.

Answer

The solution is $\left(\frac{9}{5}, \frac{4}{5}\right)$.

Steps to Solve

Identify the two equations

Let the two equations be:

\quad 2x + 3y = 6 $$

$$ 2. \quad x - y = 1 $$

Rearrange the second equation for substitution

We can express $x$ in terms of $y$ from the second equation:

$$ x = y + 1 $$

Substitute into the first equation

Now, substitute $x$ from the second equation into the first equation:

$$ 2(y + 1) + 3y = 6 $$

Simplify and solve for $y$

Distributing and combining like terms gives us:

$$ 2y + 2 + 3y = 6 \implies 5y + 2 = 6 $$

Now, subtract 2 from both sides:

$$ 5y = 4 $$

Dividing both sides by 5 gives:

$$ y = \frac{4}{5} $$

Substitute back to find $x$

Now, use the value of $y$ to find $x$ using the rearranged second equation:

$$ x = \frac{4}{5} + 1 = \frac{4}{5} + \frac{5}{5} = \frac{9}{5} $$

State the solution as an ordered pair

The solution to the system of equations is:

$$ (x, y) = \left(\frac{9}{5}, \frac{4}{5}\right) $$

The solution to the system of equations is $\left(\frac{9}{5}, \frac{4}{5}\right)$.

More Information

The solution means that the values of $x$ and $y$ satisfy both equations simultaneously. This method of substitution is often effective for solving systems of linear equations.

Tips

Forgetting to substitute correctly into the first equation.
Misapplying arithmetic when simplifying equations, especially when combining like terms.
Not checking the found values in both original equations to ensure they satisfy them.

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