91Ó°ÊÓ

When solving equations, we aim to find the values of unknown variables that make the equation true. In the given exercise, we have two linear equations involving two variables, x and y. Our goal is to determine the values of x and y that satisfy both equations simultaneously. There are various methods to solve such systems of equations. One popular method is the substitution method, which we'll explore in detail here.

The first equation given is: \[7x - 2y = -20\]
The second equation given is: \[y = x\]
We can use the second equation to substitute in the first equation. This can help us solve for one of the variables.

Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This step-by-step method of substitution helps in breaking down complex problems into simpler parts.

Algebraic substitution is a critical step in solving systems of equations. It involves replacing one variable with an equivalent expression from another equation.

In our exercise, we start by taking the second equation, \[y = x\], and substituting it into the first equation, \[7x - 2y = -20\]. This gives us: \[7x - 2(x) = -20 \]

Now, combining like terms, we get: \[7x - 2x = -20 \]
This simplifies to: \[5x = -20 \].
By solving for x, we divide both sides by 5: \[x = \frac{-20}{5} \]
which simplifies further: \[x = -4 \].

Substituting the value of x back into the second equation \[y = x\], we find: \[y = -4 \].

This step-by-step process of algebraic substitution simplifies the problem, making it easier to solve.

After solving the equations using the substitution method, it's essential to check the solutions. This ensures the values satisfy both original equations.

In our exercise, we found that \[x = -4\] and \[y = -4\].

To check our solution, substitute x and y back into the first original equation: \[7(-4) - 2(-4) = -20\].
Simplifying, we get: \[-28 + 8 = -20\].
Both sides of the equation are equal, confirming that our solution is correct.

Always verify your solutions by substituting them back into the original equations. This step reinforces your understanding and ensures accuracy. Happy learning!