Problem 4 - Elementary Algebra

Question

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair. $$ 10 x-5 y-20=0, \text { if } x=-8 $$

Step-by-Step Solution

Answer

Question: Find the value of y when x = -8 in the linear equation 10x - 5y - 20 = 0, and write the solution as an ordered pair. Answer: The solution to the given linear equation is the ordered pair (-8, -20).

Step 1: Write down the given equation and the value of x

The given equation is: $$ 10x-5y-20=0 $$ And the given value of x is: $$ x=-8 $$

Step 2: Substitute the value of x into the equation

Replace x in the equation with -8: $$ 10(-8) - 5y - 20 = 0 $$

Step 3: Simplify and solve for y

Now, perform multiplication and addition in the equation: $$ -80 - 5y - 20 = 0 $$ Combine like terms: $$ -100 - 5y = 0 $$ Now, add 100 to both sides of the equation: $$ -5y = 100 $$ Next, divide both sides of the equation by -5: $$ y = -20 $$

Step 4: Write the solution as an ordered pair

The given value of x is -8, and we found the value of y to be -20. Therefore, the solution to the given linear equation is the ordered pair: $$ (-8, -20) $$

Key Concepts

ordered pairs

In mathematics, an ordered pair is a fundamental concept used to denote a set of two elements where order is important. Imagine you are mapping locations on a grid. Each location requires two coordinates: one for the horizontal position (x), and one for the vertical position (y). Ordered pairs are usually represented as

• (x, y)

being their general form. These pairs are said to be 'ordered' because changing the order would change the information they convey.

For instance, (3, 5) is not the same as (5, 3).
In the context of linear equations in two variables, solutions are expressed as ordered pairs where the equation becomes true when these values are plugged in. Ordered pairs are essential for graphing linear equations, as each pair corresponds to a specific point on the Cartesian plane.

solving equations

Solving equations is the process of finding the unknown variable's value or values that make the equation true. In the realm of linear equations, this typically involves determining values for one or more variables that satisfy the equation. Consider a simple linear equation:

• 10x - 5y - 20 = 0

To solve this equation when x is assigned a particular value (say -8), a substitution into the equation is necessary. By inserting the value of x, the equation transforms into a simpler form:

• 10(-8) - 5y - 20 = 0

allowing us to solve for y. Solving linear equations often involves combining like terms, employing addition or subtraction, and occasionally multiplication or division.
Ultimately, solving equations is like piecing together a puzzle; each step followed systematically, narrows down to finding the precise value of unknowns.

substitution method

The substitution method is a powerful technique often used to solve systems of equations, especially linear ones. This method involves substituting a given value into one equation to find the unknown variable. It’s like filling in missing pieces with known information.
Given our equation:

• 10x - 5y - 20 = 0

we know x equals -8. So, we substitute this value in place of x.

• 10(-8) - 5y - 20 = 0

After substitution, the equation is simplified step-by-step to solve for the remaining variable y. This makes breaking down the problem easier and straightforward.
First, perform the multiplication:

• -80 - 5y - 20 = 0

Next, combine like terms:

• -100 - 5y = 0

Add 100 to both sides and divide by -5 to isolate y.
The substitution method is not only straightforward but ensures accuracy by directly replacing variables based on the provided relationships within equations. This technique is particularly useful when direct manipulation of variables can simplify a complex equation to a simpler form.