Solving Systems by Substitution

Methodology: Solving a system of linear equations using the substitution method involves expressing one variable in terms of another from one equation and substituting that expression into the second equation to produce a single-variable equation.

Step 1: Label initial equations
- Label the given system of equations as equation $(1)$ and equation $(2)$ .
Step 2: Isolate a variable
- Isolate one variable ( $x$ or $y$ ) in whichever equation is easiest to rearrange.
- Selection Criterion: Look for a variable with a coefficient of $1$ or $-1$ to make isolation straightforward.
- Designate this newly rearranged/isolated equation as equation $(3)$ .
Step 3: Substitute into the other equation
- Substitute equation $(3)$ into the remaining equation that was not manipulated in Step 2.
Step 4: Solve for the first variable
- Solve the resulting single-variable equation to determine its numerical value.
Step 5: Substitute back to determine the second variable
- Substitute the numerical value obtained in Step 4 back into equation $(3)$ to solve for the value of the remaining variable.
Step 6: State the final point of intersection
- State the final answer explicitly as an ordered pair representing the Point of Intersection $(x, y)$ .