Problem 10 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ 6 y+3=y+8 $$

Step-by-Step Solution

Answer

Question: Graph the linear equation 6y + 3 = y + 8 and describe its key features. Answer: The graph of the given linear equation is a horizontal line passing through all points with a y-coordinate of 1. The line can be represented by the equation y = 1. Its key features include: - It has a slope of 0, which indicates that it is a horizontal line. - It passes through the y-axis at the point (0, 1). - It has no x-intercept since it is parallel to the x-axis and never intersects it.

Step 1: Isolate y in the equation

To isolate y in the given equation, we will first move any terms containing y to one side of the equal sign. $$ 6y + 3 = y + 8 $$ Subtract y from both sides: $$ 5y + 3 = 8 $$ Now subtract 3 from both sides to isolate the term containing y: $$ 5y = 5 $$ Finally, divide by 5 to solve for y: $$ y = 1 $$

Step 2: Graph the line

The equation we found is y = 1, which is a horizontal line passing through all points on the Cartesian plane with a y-coordinate of 1. To graph this line, simply draw a horizontal line through the point (0,1) on the y-axis. This line represents the set of all points that satisfy the given equation.

Key Concepts

Solving Equations

Solving equations is like finding the missing piece of a puzzle. It involves figuring out the value of an unknown variable that makes an equation true. For the equation \(6y + 3 = y + 8\), our goal is to solve for \(y\). This means we need to find what number \(y\) can be to balance both sides of the equation. Here's how it works:

• Move terms around: We begin by getting all the terms with \(y\) on one side. So, we subtract \(y\) from both sides: \(6y + 3 - y = 8\).


• Simplify: Combine like terms to make \(5y + 3 = 8\) simpler.


• Isolate \(y\): Subtract 3 from both sides to get \(5y = 5\).


• Find \(y\): Divide by 5, resulting in \(y = 1\).

This result tells us that when \(y\) is 1, the equation holds true.

Graphing Linear Equations

Graphing linear equations gives us a visual picture of solutions. When we graph \(y = 1\), we can better understand what the equation represents. To do this, follow these steps:

• Since \(y = 1\) tells us \(y\) is always 1, we draw a horizontal line. This line crosses the y-axis at the point (0,1).


• This line doesn't tilt up or down, because \(y\) is fixed. It's the same for all values of \(x\).


• Think of this as a flat surface where every point on the line has a y-value of 1.

This helps us see that any point on this line is a solution to the equation, not just \(x = 0\). Any \((x, 1)\) works.

Isolating Variables

Isolating variables is an essential part of solving equations. It helps us "unlock" the value of the unknown. To isolate means to get the variable by itself on one side of the equation. Consider \(6y + 3 = y + 8\):

• First action: Remove \(y\) from the right side by subtracting it from both sides. This starts isolating \(y\).


• Next: Simplify by subtracting numbers without variables, like moving the 3 from the left side by subtracting 3 from both sides: \(5y = 5\).


• Final step: Divide both sides by the number next to \(y\) (which is 5) to get \(y = 1\).

This step-by-step approach helps us systematically find what \(y\) must be to satisfy the equation.