Lesson 6-3: Solve Systems of Equations by Substitution

Property

To use substitution, you must isolate one variable (make its coefficient 1 or -1) on one side of the equation. To minimize algebraic work, strategically select a variable that already has a coefficient of 1 or -1. If an equation contains fractions, multiply every term on both sides by the least common denominator (LCD) to eliminate them before isolating a variable.

Examples

• To solve $3x + y = 10$ for $y$, subtract $3x$ from both sides, which gives you $y = 10 - 3x$.
• In the system $2x + y = 5$ and $3x - 4y = 10$, the easiest variable to isolate is $y$ in the first equation because its coefficient is 1. Subtracting $2x$ from both sides gives $y = -2x + 5$, completely avoiding fractions.
• Given the equation $\frac{1}{2}x + \frac{2}{3}y = 4$, multiply every term by the LCD, which is 6, to get $3x + 4y = 24$.

Explanation

Choosing the right variable to isolate can save time and prevent calculation errors. Always scan both equations for a variable with a coefficient of 1 or -1, as isolating it will prevent you from having to divide and create messy fractions. If an equation already contains fractions, you can eliminate them by multiplying every term on both sides by the least common denominator (LCD). These strategies minimize complex algebraic work and make the substitution step much smoother.

Property

To solve a system by substitution, follow these steps:

1. Solve one of the equations for either variable.
2. Substitute the expression from Step 1 into the other equation.
3. Solve the resulting equation.
4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
5. Write the solution as an ordered pair and check that it is a solution to both original equations.

Examples

• Solve the system $y = x + 3$ and $3x + 2y = 19$. Substitute $x+3$ for $y$ in the second equation: $3x + 2(x+3) = 19$. This simplifies to $5x+6=19$, so $5x=13$ and $x=\frac{13}{5}$. Then $y = \frac{13}{5} + 3 = \frac{28}{5}$. The solution is $(\frac{13}{5}, \frac{28}{5})$.
• Solve the system $2x - y = 8$ and $x + 3y = 11$. From the first equation, solve for $y$: $y = 2x - 8$. Substitute this into the second equation: $x + 3(2x-8) = 11$. This gives $7x-24=11$, so $7x=35$ and $x=5$. Then $y=2(5)-8=2$, making the solution $(5, 2)$.

Explanation

This method simplifies a two-variable system into a single-variable equation. By isolating a variable in one equation and plugging its expression into the other, you can solve for one variable and then use that value to find the second.