Problem 26 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$-\frac{2}{3} x-6=-4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\). This solution is confirmed graphically.

Step 1: Isolate x Term

The first step is to isolate the x term on one side of the equation. Add 6 to both sides of the equation; that will result in the new equation: \(-\frac{2}{3}x = -4 + 6 = 2\).

Step 2: Solve for x

Now \(x\) term can be isolated by multiplying both sides of the equation by \(-\frac{3}{2}\). The equation becomes: \(x = 2 * -\frac{3}{2} = -3\).

Step 3: Graphical Check

To verify the solution graphically, draw a graph of the function \(y = -\frac{2}{3}x - 6\). Locate the point where the curve crosses the x-axis. This is the solution. -3 should indeed be the x-coordinate where this curve crosses the x-axis, which verifies the correctness of the obtained solution graphically.

Key Concepts

Solving Equations

Solving algebraic equations involves finding the value of the unknown variable that makes the equation true. The given equation is \[-\frac{2}{3}x - 6 = -4\].
To solve it, we aim to isolate the variable \(x\) on one side of the equation. Here are some steps to follow:

• Add or subtract numbers to both sides to eliminate constants from the side containing the variable. In this equation, adding 6 to both sides removes the -6.
• Simplify both sides of the equation if needed. Once simplified, the equation might look like \(-\frac{2}{3}x = 2\).
• To solve for \(x\), multiply both sides by the reciprocal of the fraction coefficient of \(x\). Since the coefficient is \(-\frac{2}{3}\), we multiply by \(-\frac{3}{2}\) to isolate \(x\).

At this stage, you can confirm that \(x = -3\). Through these steps, we can effectively find the solution for \(x\) in any similar linear equation.

Graphing Functions

Graphing functions is a visual method to understand and solve equations. By transforming the equation \(-\frac{2}{3}x - 6 = y\),
we can graph this line in a coordinate plane. Here's how visualization assists:

• Set the equation in the form \(y = mx + b\), where \(m\) is the slope. Our equation \(-\frac{2}{3}x - 6 = y\) is already in this form.
• The slope \(m\) is \(-\frac{2}{3}\). This means as \(x\) increases by 1, \(y\) decreases by \(\frac{2}{3}\).
• The y-intercept \(b\) is -6, where the line crosses the y-axis.

To graph it:- Start at the y-intercept (0, -6) on the graph.- From there use the slope to find another point. Go down 2 units and over 3 units with each step to represent the decrease in y with an increase in x.
Plotting these points will give you a line that, when extended, crosses the x-axis at -3. This crossing point visually confirms our solution from solving the equation algebraically, showing how graphing functions helps in checking solutions.

Linear Equations

Linear equations form the backbone of algebra and are characterized by lines when graphed. The equation \(-\frac{2}{3}x - 6 = -4\) represents a linear equation which is quite straightforward once you understand its components:

• Linear equations typically include one variable with a constant rate of change, which is the slope.
• The form \(y = mx + b\) represents all linear equations, where \(m\) denotes the slope and \(b\) is the y-intercept.

Understanding the structure allows us to comprehend:- The rate at which two variables relate, shown by the slope \(m\).- Where the line intersects the y-axis, evident from the intercept \(b\).
Linear equations are simple to solve and graph, making them a fundamental concept in algebra. They reveal relationships linearly and are a building block for more complex math topics.