Question

The larger of two numbers exceeds twice the smaller number by 8 . The sum of twice the larger and 3 times the smaller number is \(65 .\) If \(x\) is the smaller number, which equation below determines the correct value of \(x ?\) F. \(3(2 x+8)+2 x=65\) G. \(3(2 x+8)+2 x=65\) H. \(\quad(4 x+8)+3 x=65\) J. \(2(2 x+8)+3 x=65\) K. \(2(2 x-8)+3 x=65\)

Short answer

Answer: J. \(2(2x+8)+3x=65\)

Step-by-step solution

Translate the first statement into an equation

The larger of two numbers exceeds twice the smaller number by 8. If we let the larger number be L and the smaller number be x, we can write an equation from this statement as: L = 2x + 8

Translate the second statement into an equation

The sum of twice the larger and 3 times the smaller number is 65. Using the variables L and x as defined earlier, we can write this statement as: 2L + 3x = 65

Substitute L from Step 1 into the equation from Step 2

Now, we have a system of two equations with two variables. We can use substitution method to solve for x. Substitute the value of L (2x + 8) from the first equation into the second equation: 2(2x + 8) + 3x = 65

Compare the resulted equation to the choices

We have found the following equation that determines the correct value of x: 2(2x + 8) + 3x = 65 Comparing this equation to the given choices, we can see that it matches option J: J. \(2(2x+8)+3x=65\) So, the correct answer is J.

Key concepts

Algebraic Equations

An algebraic equation is a statement of equality between two expressions composed of variables and constants. The primary goal in algebra is to find the value of the unknown variables by solving the equations. To solve the equation, one must perform operations that will simplify the equation and isolate the variable, eventually revealing its value.

For instance, if the equation is presented as 'the larger of two numbers exceeds twice the smaller number by 8,' you are tasked with translating this word problem into an algebraic equation. This means identifying a variable (often 'x') to represent an unknown value and constructing an equation that correctly depicts the relationship described. In the given exercise, a student needs to represent the larger number as a function of the smaller one, leading to an equation like 'L = 2x + 8,' where 'L' is the larger number, 'x' is the smaller number, and '8' is the constant value by which the larger number exceeds twice the smaller one.

Substitution Method

The substitution method is one of the techniques used to solve systems of equations. It involves replacing one variable with its equivalent expression from another equation within a system, thus reducing the number of unknown variables.

Let's dig deeper into this method using the problem we have. With two equations, 'L = 2x + 8' and '2L + 3x = 65,' the substitution method comes into play by expressing one variable—in this case, 'L'—in terms of 'x' from the first equation. Then you can substitute '2x + 8' in the place of 'L' in the second equation. This action transforms the system of equations into a single equation in one variable:2(2x + 8) + 3x = 65.

Once substituted, you can solve for 'x' directly, bringing you closer to answering the problem at hand. The substitution method is a brilliant tool for tackling algebra problems as it simplifies equations into more manageable forms.

System of Equations

A system of equations consists of two or more equations with the same set of variables. Solving a system of equations means finding the values of the variables that make all the equations true simultaneously. Most often, these systems are linear, but they can also include non-linear equations.

In the context of our example, we are dealing with a system involving two linear equations:

Equation 1: L = 2x + 8

Equation 2: 2L + 3x = 65

The goal is to find a common solution that satisfies both equations. In addition to substitution, other methods like elimination or graphing can be used depending on the specifics of the problem. Systems of equations are foundational in algebra, often representing real-world problems where multiple conditions must be met simultaneously, as seen with the relationship between two numbers in our exercise.