Substitution method
Solve systems of equations using the substitution method: isolate one variable in one equation, substitute the expression into the other, then solve the resulting single-variable equation.
Key Concepts
Property The substitution method is a method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation.
Solve $\begin{cases} y = 2x + 1 \\ 2y x = 6 \end{cases}$. Substitute $(2x+1)$ for y in the second equation: $2(2x+1) x = 6$, which simplifies to $x = \frac{8}{3}$. Solve $\begin{cases} x = 3y 1 \\ 2x + 4y = 16 \end{cases}$. Substitute $(3y 1)$ for x in the second equation: $2(3y 1) + 4y = 16$, which simplifies to $y = \frac{9}{5}$.
Think of substitution as a clever disguise for a variable. First, you solve one equation to find out what a variable (like 'y') is equal to. Then, you sneak that expression into the other equation, replacing the original 'y'. This move eliminates one variable, letting you easily solve for the other and crack the system's code.
Common Questions
What are the steps for the substitution method?
Choose one equation and solve it for one variable in terms of the other. Substitute that expression into the second equation. Solve the resulting one-variable equation. Substitute the found value back into the first equation to find the other variable. Check both values in the original equations.
When is substitution more efficient than elimination?
Substitution is most efficient when one equation already has a variable isolated or has a coefficient of 1 or -1, making isolation easy. For example, if one equation reads y=2x+3, substitution into the other equation is immediate without any additional manipulation.
What does it mean if substitution leads to a true statement like 0=0?
If all variables cancel and you get a true statement like 0=0 or 5=5, the system is dependent and has infinitely many solutions. The two equations describe the same line. If you get a false statement like 0=4, the system is inconsistent with no solution.