The first step in solving a system of equations using the substitution method is to solve one equation for one of the variables.
Based on the provided reference information, the initial action required when applying the substitution method is to isolate a single variable in either of the equations presented in the system. This means rearranging one of the equations so that it is in the form "variable = expression" or "expression = variable".
Understanding the First Step
This foundational step is crucial because it allows you to express the value of one variable in terms of the other variable. Once you have one variable isolated, you can then substitute that expression into the other equation in the system.
How to Perform the First Step
- Choose an equation: Look at both equations in the system.
- Choose a variable: Decide which variable (like $x$ or $y$) you want to isolate within the chosen equation. Often, it's easiest to choose a variable that already has a coefficient of 1 or -1.
- Isolate the variable: Use algebraic operations (adding, subtracting, multiplying, dividing) to get the chosen variable by itself on one side of the equation.
Example:
Consider the system:
$x + y = 5$
$2x - y = 1$
To perform the first step, you could choose the first equation ($x + y = 5$) and isolate $x$:
$x + y - y = 5 - y$
$x = 5 - y$
Alternatively, you could choose the first equation and isolate $y$:
$x + y - x = 5 - x$
$y = 5 - x$
Or, you could choose the second equation and isolate $y$ (since it has a coefficient of -1, isolating $-y$ first might be simple):
$2x - y = 1$
$2x - 1 = y$
Any of these resulting expressions ($x = 5 - y$, $y = 5 - x$, or $y = 2x - 1$) represent completing the first step.
The Full Substitution Process
To put the first step into the context of the entire method, the referenced steps are:
- Solve one equation for one of the variables. (As explained above)
- Substitute (plug-in) this expression into the other equation and solve.
- Resubstitute the value into the original equation to find the corresponding variable.
By successfully completing the first step, you prepare the system for the subsequent substitution and solving phases.
