Math 1050: A Visual Journey Through Algebra

Welcome to the foundations of algebra! Today, we'll explore how numbers and variables work together.In algebra, we use letters called variables to represent unknown numbers.Let's look at a real-world example with shopping. Apples cost 2 dollars each, and bananas cost 3 dollars each.If we want to calculate the total cost for any number of apples and bananas, we can use variables. Let x be the number of apples and y be the number of bananas.Let's try different combinations to see how our expression works.Now, let's look at another example using distance and perimeter.In this rectangle, we use l for length and w for width. The perimeter is the distance around the entire shape.The formula for perimeter is two times the length plus two times the width.Let's try different values for length and width to see how the perimeter changes.Now that we understand variables, we're ready to learn how to solve equations!When solving equations, we must maintain balance by performing the same operations on both sides.Let's solve this equation step by step, starting with subtracting 5 from both sides to isolate the term with x.After simplifying, we have two x equals eight.Now we divide both sides by two to isolate x.Simplifying gives us our solution: x equals four.It's crucial to check our solution by plugging the value back into the original equation.Let's substitute x equals 4 into two x plus 5 equals 13.First multiply: two times four is eight.Then add five: eight plus five equals thirteen. Our solution is correct!Let's review some common mistakes to avoid when solving equations.Always remember to apply operations to both sides, simplify each step, and show all your work clearly.Now try solving this equation on your own: three x minus seven equals fourteen. Remember to follow the same steps we just learned.On our coordinate plane, we can visualize linear equations by plotting points and drawing lines.Let's start with the equation y equals 2x plus 1.To graph this equation, we can start by plotting some points. Let's find coordinates by plugging in x values.To find the slope between two points, we look at the rise over run. Here, when we move right 2 units, we go up 4 units.Now we can draw our line through these points. Notice how it passes through each point we plotted.Let's see how changing the values in our equation affects the line on our graph.When we increase the slope to 3, our line becomes steeper.And when we decrease the slope to 1, the line becomes flatter.The y-intercept determines where our line crosses the y-axis. Let's move it up to positive 3.And now let's move it down to negative 1.Parallel lines have the same slope. Here's another line with slope 1.Perpendicular lines have slopes that are negative reciprocals of each other. This line with slope negative 1 is perpendicular to our lines with slope 1.