Problem 20 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ -20 y+14 x=1, \text { if } x=8 $$

Step-by-Step Solution

Answer

Answer: y = 111/20

Step 1: Substitute the value of x into the equation

Replace x with 8 in the given equation: $$ -20y + 14 (8) = 1 $$

Step 2: Simplify the equation

Calculate 14 times 8 and simplify the equation: $$ -20y + 112 = 1 $$

Step 3: Solve for y

To solve for y, first subtract 112 from both sides of the equation: $$ -20y = -111 $$ Now, divide both sides by -20: $$ y = \frac{-111}{-20} $$

Step 4: Simplify the fraction for y

Simplify the fraction to get the value of y: $$ y = \frac{111}{20} $$ The solution for the given linear equation when x=8 is: $$ y = \frac{111}{20} $$

Key Concepts

Linear Equations in Two Variables

Linear equations in two variables are mathematical expressions that represent a straight line when graphed on a coordinate plane. These equations have the general form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are the variables that can take any value. Here, \( a \) and \( b \) cannot both be zero since at least one variable must be present. The solution to such an equation is a pair of numbers that make the equation true when substituted for \( x \) and \( y \).
To find the solution, one can use several methods; plotting graphs, substitution, elimination, or matrices. The best method to use often depends on the specific problem and the numbers involved. In the case of the given textbook example, where the equation is \( -20y + 14x = 1 \) and \( x \) is already given as 8, the substitution method becomes the most efficient way to find the value of \( y \).

Substitution Method

The substitution method is a technique used to find the solution to systems of linear equations. It involves isolating one variable in one equation and then substituting that expression into the other equation. This method is particularly useful when one of the variables has a coefficient of 1 or when a variable is already given a value, as in the exercise provided.
In the example, you are given \( x = 8 \) and need to solve for \( y \) in the equation \( -20y + 14x = 1 \). The substitution process includes replacing \( x \) with 8 and then proceeding to solve the resulting equation for \( y \). This method simplifies the problem by reducing it to a one-variable equation, which is generally easier to solve. The key to mastering this method is to ensure you perform the substitution correctly and then apply basic algebra to solve for the remaining variable.

Simplifying Equations

Simplifying equations means to perform algebraic manipulations that make them easier to solve. These manipulations include operations like adding or subtracting the same number from both sides, multiplying or dividing both sides by the same number, and combining like terms. The goal is to isolate the variable you are solving for, making it the subject of the equation.
In the exercise's final steps, you encounter the process of simplifying the equation via subtraction and division. After substituting \( x = 8 \), you multiply to find \( -20y + 112 = 1 \). The next step is to subtract 112 from both sides to isolate the terms containing \( y \), resulting in \( -20y = -111 \). Lastly, you divide both sides by -20, simplifying to \( y = \frac{111}{20} \), which gives you the value of \( y \). Understanding how to simplify correctly is crucial as it allows you to solve equations efficiently and accurately.