Problem 5 - Elementary Algebra

Question

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair. $$ 3 a+2 b+6=0, \text { if } a=-1 $$

Step-by-Step Solution

Answer

Answer: The solution of the linear equation when a = -1 is (-1, -3/2).

Step 1: Substitute the given value of a in the equation

We are given that \(a = -1\). We substitute this value in the given equation: $$ 3(-1) + 2b + 6 = 0 $$

Step 2: Solve the equation for b

Now we have an equation with only one variable, \(b\). Let's solve it: $$ -3 + 2b + 6 = 0 $$ $$ 2b + 3 = 0 $$ $$ 2b = -3 $$ $$ b = -\frac{3}{2} $$

Step 3: Write the solution as an ordered pair

Now that we have the value of \(b\), we can write the solution as an ordered pair \((a, b)\). Since \(a = -1\) and \(b = -\frac{3}{2}\), the solution is: $$ (-1, -\frac{3}{2}) $$

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to locate a point in a coordinate plane. An ordered pair is written in a specific sequence as

• \((x, y)\) in two-dimensional space
• \((a, b)\) is another example, where 'a' and 'b' are specific numbers

The first component represents the x-coordinate or the horizontal position. The second component represents the y-coordinate or the vertical position.
Ordered pairs are especially useful in graphing solutions to equations involving two variables.
In the original exercise, the solution of the equation is written as an ordered pair \((-1, -\frac{3}{2})\), indicating that when \(a = -1\), the value of \(b\) is \(-\frac{3}{2}\).
This notation helps in visually understanding the relationship between the two variables.

Substitution Method

The substitution method is a technique used to solve systems of equations. It's a strategy where you solve one equation for one variable, and then substitute that expression into another equation. In this specific exercise:

• We have the equation \(3a + 2b + 6 = 0\)
• Given that \(a = -1\), we've substituted \(-1\) into the equation for \(a\)

This simplifies the problem to one variable, which is much easier to solve.
Substitution transforms a complex problem into a manageable one.
It involves clarity about the role of each symbol in the equation, making it straightforward to isolate and solve for the desired unknown.

Solving Equations

Solving equations is a critical skill in algebra, often requiring a step-by-step approach. In the given exercise, we start by substituting known variables:

• The equation becomes \(3(-1) + 2b + 6 = 0\)
• Simplify the equation: \(-3 + 2b + 6 = 0\)
• Re-arrange to isolate \(b\): \(2b + 3 = 0\)
• Subtract 3 from both sides: \(2b = -3\)
• Divide both sides by 2: \(b = -\frac{3}{2}\)

Each step brings us closer to the solution, revealing the value of the variable in question.
Following these steps accurately ensures that the solution is correct and comprehensive.
Practicing these methods improves one's skills in algebra and develops the analytical abilities needed to tackle more complex mathematics.