Problem 32 - Algebra 1

Question

Solve the equation graphically. Check your solution algebraically. $$6 x+9=3$$

Step-by-Step Solution

Answer

The solution to the equation \(6x+9 = 3\) is \(x = -1\), as confirmed both graphically (where the graph of \(y = 6x+9\) intersects the x-axis), and algebraically (by substitution into the original equation).

Step 1: Rearrange the equation

First, we need to rearrange the equation to be able to graph it. Let's start with \(6x+9=3\). We isolate x by subtracting 9 from both sides to get \(6x = -6\), then divide by 6 to get \(x = -1\).

Step 2: Graph the equation

Next, we graph the equation \(y = 6x+9\). The graph will intersect the x-axis at our solution. In this case, our solution is \(x = -1\), and the graph intersects the x-axis at \(x = -1\).

Step 3: Check the solution algebraically

Finally, we verify the solution by substituting \(x = -1\) into the original equation. Doing so yields \(6*(-1) + 9 = 3\), which is the same as our original equation, verifying the correctness of our solution.

Key Concepts

Graphical Solution

A graphical solution involves using a graph to find the value of the variable that satisfies the equation. For the linear equation \(6x + 9 = 3\), we can represent it as a function \(y = 6x + 9\). By plotting this function on a graph, we visualize how \(y\) changes with respect to \(x\). The x-coordinate where the graph intersects the x-axis (where \(y = 0\)) represents the solution to the equation.

To determine this graphically, follow these steps:

• First, rearrange the equation in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In this case, the equation is already in the form: \(y = 6x + 9\).
• Plot the y-intercept \((0, 9)\) on the graph.
• Use the slope to find another point. The slope of 6 means that for every 1 unit increase in \(x\), \(y\) increases by 6 units.
• Draw the line through these points.

This line will cross the x-axis at \((-1, 0)\), indicating that \(x = -1\) is the solution.

Graphical methods are useful when equations are difficult to solve algebraically or when you desire a visual understanding of the solution.

Algebraic Verification

Algebraic verification is the process of confirming the solution obtained through graphical methods or other techniques is correct. It ensures that the solution satisfies the original equation.
For our example, check the solution \(x = -1\) by plugging this value back into the original equation:

• Start with the equation: \(6x + 9 = 3\).
• Substitute \(-1\) for \(x\): \(6(-1) + 9 = 3\).
• This simplifies to: \(-6 + 9 = 3\).
• Finally, verify: \(3 = 3\), which is true.

Since both sides of the equation are equal, our solution \(x = -1\) is correct. This method is crucial for double-checking solutions derived from graphs or other methods, especially if the visual representation could be imprecise or misinterpreted.

Linear Equations

Linear equations are algebraic expressions that describe a line on a graph. They take the general form \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.

Here's what makes linear equations unique and easy to understand:

• The graph of a linear equation is always a straight line.
• The slope \(m\) tells you how steep the line is and in which direction it goes.
• The y-intercept \(b\) shows where the line crosses the y-axis.

Solving a linear equation involves finding the value of \(x\) that makes the equation true. This solution is the x-coordinate of the point where the line intersects the x-axis.

In our exercise \(6x + 9 = 3\), which simplifies to \(x = -1\), the equation tells us how \(y\) responds for each unit change in \(x\). By understanding the structure and behavior of linear equations, you can tackle a wide array of mathematical problems with confidence and clarity.