91Ó°ÊÓ

The Substitution Method is a way to solve systems of linear equations. It involves isolating one variable in one of the equations and then substituting that expression into the other equation. This leads to a single equation with one variable, which is easier to solve.

For instance, in the given problem, we start with the system:
\(x + 3y = 1 \),
\(3x + 5y = -5 \).
By solving the first equation for \(x\), we get: \[ x = 1 - 3y \]
We then substitute this expression into the second equation. Doing so simplifies the problem to one equation with one variable. This manageable equation is then solved to find the value of \(y \). Once we have \(y \), we substitute it back into the expression for \(x \).

Using the Substitution Method, you're systematically breaking down the problem, making it easier to get to the solution.

Linear equations are algebraic expressions that represent straight lines when graphed. They have variables raised to the power of one and follow the form \(ax + by = c\).

In our problem, we have two linear equations: \[ x + 3y = 1 \] \[ 3x + 5y = -5 \]
Each equation represents a line in a coordinate plane. The solution to the system of these linear equations is the point where these lines intersect. By knowing how to represent one variable in terms of another, we can find that point algebraically.

Linear equations are fundamental in various fields, including physics, economics, and statistics, because they describe consistent relationships between variables.

Solving systems of equations using the Substitution Method follows clear algebraic steps. These steps ensure you can find the values of the variables systematically:

• Solve one equation for one variable: Choose one of the equations and solve for one variable. In our case, we solved \(x + 3y = 1\) for \(x\), getting \[ x = 1 - 3y \].
• Substitute the expression: Substitute the expression derived in the first step into the other equation. We substituted \(x = 1 - 3y\) in \(3x + 5y = -5\): \[ 3(1 - 3y) + 5y = -5 \].
• Simplify and solve: Simplify the resulting equation and solve for the second variable, \(y\). For us, it simplifies to: \[ 3 - 9y + 5y = -5\], then solving gives \[ y = 2 \].
• Substitute back: Substitute the value of \(y\) back into the expression for \(x\) to find its value: \[ x = 1 - 3(2) = -5 \].
• Check the solution: Finally, ensure the solution satisfies both original equations. Substituting \(x = -5\) and \(y = 2\) into both equations confirmed their correctness.
  
  By keeping these algebraic steps in mind, solving systems of equations by substitution becomes a straightforward process.