Question

Determine whether the given value of \(x\) is a solution to the equation. \(3(x-2)+4 x-1=x-1 ; x=1\)

Short answer

Yes, the given value \(x = 1\) is a solution to the equation.

Step-by-step solution

Substitute the given value of \(x\)

Replace all instances of \(x\) in the equation with the given value \(x = 1\).So the equation \(3(x-2) + 4x - 1 = x - 1\) becomes \(3(1 - 2) + 4*1 - 1 = 1 - 1\).

Simplify both sides

By simplifying each side of the equation we get:LHS: \(3*(-1) + 4 - 1 = -3 + 4 - 1 = 0\),RHS: \(1 - 1 = 0\).

Check if LHS equals RHS

Both the LHS and the RHS are equal to 0. So, the given value \(x = 1\) is indeed a solution to the equation.

Key concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to solve equations. It involves replacing variables with their respective numerical values or other algebraic expressions. This method is particularly useful when you have a specific value for a variable, as is often the case when testing whether the value is a solution to an equation.

In the context of the given exercise, the substitution method was applied by taking the given value of x, which is 1, and replacing all occurrences of x in the equation with this number. This step transforms the equation into a numerical expression, which can then be simplified through standard arithmetic operations.

The substitution method not only helps in verifying solutions but also lays the foundation for more advanced algebraic techniques such as solving systems of equations, where one equation can be solved for one variable and then substituted into other equations.

Algebraic Expressions

Understanding algebraic expressions is crucial in the realm of algebra. These expressions consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. An expression can represent a particular value once the variables are replaced by specific numbers, a process often seen in the substitution method explained above.

For the equation in the original exercise, 3(x-2) + 4x - 1 is an algebraic expression on the left-hand side (LHS), and x - 1 on the right-hand side (RHS). Throughout the solution process, it's vital to manipulate these expressions correctly by following the conventional order of operations: parentheses, exponents, multiplication and division, and finally addition and subtraction.

Creating Equivalent Expressions

It's also important to recognize that equivalent expressions can be created by combining like terms or using the distributive property. For instance, in the given exercise, the expression 3(x-2) can be expanded to 3x - 6, which can then be combined with 4x yielding 7x - 6, an equivalent expression that simplifies the process of solving the equation.

Equation Simplification

The process of equation simplification involves reducing an equation to its simplest form, making it easier to solve or interpret. This typically includes combining like terms, which are terms that have the same variable and exponent, and carrying out any required arithmetic operations to minimize the number of terms.

In our exercise, once the substitution of x = 1 was made, the equation simplification involved performing arithmetic to combine and reduce the terms on both sides, turning them into a single numerical value. The left-hand side (LHS) involved calculating 3(1-2) + 4*1 - 1, and the right-hand side (RHS) is the simple operation of 1 - 1.

Order of Operations

Always remember to follow the order of operations (PEMDAS/BODMAS) during simplification. In our example, the subtraction within the parentheses is performed first, followed by the multiplication and finally the addition and subtraction. Simplifying both sides led to the confirmation that the given value of x is indeed a solution, as both sides simplified to zero, creating a true statement when each side of the equation was compared.