Determining Solutions by Substitution
Determining whether a value is a solution to an equation is done by substituting the value into the equation and checking if it makes a true statement. This verification skill is taught in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations. It is a foundational technique for checking answers after solving any algebraic equation.
Key Concepts
Property A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. To determine whether a number is a solution to an equation:.
Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
Examples Is $x=5$ a solution to the equation $3x 5 = 10$? We substitute $3(5) 5 = 15 5 = 10$. Since $10 = 10$, it is a solution. Is $y= 2$ a solution to the equation $4y + 9 = 2y$? We substitute $4( 2) + 9 = 8 + 9 = 1$ on the left, and $2( 2) = 4$ on the right. Since $1 \neq 4$, it is not a solution. Is $a = \frac{1}{3}$ a solution to the equation $9a + 2 = 5$? We substitute $9(\frac{1}{3}) + 2 = 3 + 2 = 5$. Since $5=5$, it is a solution.
Common Questions
How do you check if a value is a solution to an equation?
Substitute the value in place of the variable. If both sides of the equation are equal after simplifying, it is a solution.
What does it mean for a value to be a solution?
A solution is a value that makes the equation a true statement when substituted for the variable.
Where is determining solutions by substitution taught in Openstax?
This is in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations.
What if both sides are not equal after substitution?
If the two sides are not equal, the value is not a solution to the equation.
Why is checking solutions by substitution important?
Substitution verifies your work and catches arithmetic errors, and it identifies extraneous solutions introduced during solving.