Isolating a Variable for Substitution
Isolating a Variable for Substitution teaches a strategic step in solving systems of equations by substitution. From OpenStax Prealgebra 2E, the goal is to get one variable alone (coefficient of 1 or −1) before substituting into the other equation. In 3x + y = 10, subtracting 3x isolates y directly: y = 10 − 3x. When choosing which variable to isolate, pick one that already has a coefficient of 1 or −1 to minimize algebraic work. If fractions appear, multiply all terms by the LCD first to clear them.
Key Concepts
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.
Common Questions
What does isolating a variable mean in substitution?
It means rearranging one equation so a single variable is alone on one side with coefficient 1, ready to be substituted into the other equation.
How do you isolate y in 3x + y = 10?
Subtract 3x from both sides: y = 10 − 3x.
Which variable should you isolate when solving by substitution?
Choose the variable with a coefficient of 1 or −1, since it requires the least work to isolate — no division needed.
How do you handle fractions when isolating a variable?
Multiply every term on both sides by the LCD to clear all fractions before isolating the variable.
In the system 2x + y = 5 and 3x − 4y = 10, which variable is easiest to isolate?
y in the first equation, because its coefficient is already 1. Subtract 2x to get y = 5 − 2x.
What is the next step after isolating a variable?
Substitute the isolated expression into the other equation in place of that variable, then solve for the remaining unknown.