Problem 27 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF EQUATIONS. Check to see if the given value of the variable is or is not a solution of the equation. $$ 5 r-10=11 ; r=5 $$

Step-by-Step Solution

Answer

\(r = 5\) is not a solution to the equation \(5r - 10 = 11\).

Step 1: Understanding the equation

The equation given in this exercise is \(5r - 10 = 11\). This is a linear equation in one variable, \(r\). Given \(r = 5\), our task is to substitute this value in place of \(r\) in the given equation and to check whether it satisfies the equation.

Step 2: Substituting the value of the variable

Substitute \(r = 5\) into the equation, which gives \(5(5) - 10 = 11\). This simplifies to \(25 - 10 = 11\).

Step 3: Verifying the solution

Simplify the left hand side of the equation, which gives \(15 = 11\). In this case, the left hand side does not equal the right hand side, so \(r = 5\) is not a solution to the equation \(5r - 10 = 11\).

Key Concepts

Linear Equations

Linear equations form the foundational building blocks of algebra and are incredibly important in understanding mathematical relationships. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The standard form is typically expressed as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

• The key feature of a linear equation is that it graphs as a straight line on a coordinate plane.
• It usually involves operations like addition, subtraction, multiplication, or division, but not exponents or roots with the variable.

To better understand linear equations, let's consider the equation provided in the exercise: \(5r - 10 = 11\). Here, \(r\) is the variable, and \(5\) and \(-10\) are constants. To find the value of \(r\), which makes the equation true, we rely on algebraic techniques like substitution.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of a variable that makes an equation true. The process involves manipulating the equation using various algebraic techniques until the variable is isolated.

• First, you identify the variable you need to solve for, which in our exercise is \(r\).
• Next, you systematically eliminate other terms from the equation step-by-step to isolate the variable.

In our given exercise, we were tasked with checking for the solution by substituting a value for \(r\) rather than isolating the variable through manipulation. This is a variation of solving equations where the focus is on verifying solutions, which can be particularly useful in checking whether a given number meets the criteria of the equation or not.

Substitution Method

The substitution method is a powerful algebraic tool used to solve equations, particularly useful in systems of equations but also applicable in single equations for verifying or determining solutions. The idea is straightforward:

• You substitute a given value for a variable in the equation and simplify to see if the equality holds.
• This method can be applied to any part of an equation to simplify it or verify if a certain value is indeed a solution.

In our exercise, by substituting \(r = 5\) into the equation \(5r - 10 = 11\), we directly tested this value for feasibility. The resultant equation \(25 - 10 = 11\) simplfied to \(15 = 11\), indicating that \(r = 5\) is not the correct solution. The substitution method thus helps confirm whether a proposed solution is valid, making it an essential technique for students to master when learning to solve equations.