Problem 79 Solve the system of equations \(... [FREE SOLUTION] (2024)

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The substitution method is a technique used to solve systems of equations. It involves solving one equation for a specific variable and then substituting that expression into the other equation. It simplifies a system of linear equations to a single equation. For example, we first solved for \(x\) in the second equation:

• \(x + 4y = -1\)
• \(x = -1 - 4y\)

We substitute this into the first equation:

• \(2(-1 - 4y) - 3y = 20\)

This method is great for systems where you can easily isolate a variable and substitute it into the other equation.

Linear equations are equations of the first degree. This means that the variables are not raised to any power higher than one. They can be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. The given system of equations:

• \(2x - 3y = 20\)
• \( x + 4y = -1\)

Both of these equations are linear. Linear equations can be graphed as straight lines on a Cartesian plane. The solution to a system of two linear equations is the point where the two lines intersect.

Variables in an equation represent unknowns that we need to solve for. In the given system, \(x\) and \(y\) are the variables. Each variable can take different values depending on the equations they are part of. Here is a step-by-step view to understand how variables were managed:

• We first solved the second equation for \(x\), making \(x\) the subject: \(x = -1 - 4y\)
• Next, we substituted this expression into the first equation to solve for \(y\): \(-11y = 22\)
• Once we had \(y\), we substituted \(y\) back into the equation solved for \(x\) to find its value: \(x = 7\)

Variables are essential components in algebra as they allow us to solve for unknown quantities in equation forms.