Master the algebraic substitution method to solve linear equations exactly.
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The limitation of graphing non-integral coordinates.
Graphing linear equations is visual and intuitive, but it is not always perfectly precise. When the lines intersect at non-integral coordinates like or fractions like , reading the exact point directly from a graph is prone to errors.
The 3 core steps of substitution.
Solve a straightforward system by substitution.
Problem. Solve the following pair of equations by the substitution method:
Substitution where variables cancel leaving a true statement.
Problem. Find the cost of a pencil and an eraser using the following pair of equations:
Substitution where variables cancel leaving a false statement.
Consider the system of linear equations and . We express in terms of from the first equation to get . Now, we substitute this value of into the second equation to get . Expanding this equation yields . After combining like terms, we arrive at the equation , which is a false statement. Because the variables are eliminated and the resulting statement is false, the equations do not have a . Consequently, the rails represented by these lines are parallel and will never cross.