Question

Solve the equation graphically. Check your answer algebraically. $$5 x+6=-9$$

Short answer

The root of the equation \(5x + 6 = -9\) is \(x = -3\), which should be the x-coordinate when the graphical line crosses \(y = -9\) on the y-axis.

Step-by-step solution

Graphical Solution

Start by setting the function \(y = 5x+6\) and then sketch this on a 2D Cartesian plane. Since this is a straight line, only two points are needed to sketch the line. A commonly chosen first point is when \(x = 0\), that implies \(y = 6\). For the second point, when \(y = 0\), the x-value can be determined as follows: \[0=5x+6 \Rightarrow x = -\frac{6}{5} = -1.2\]So, the two points are \((0, 6)\) and \((-1.2, 0)\). Draw a line passing through these two points. After drawing the line, we need to find the x-coordinate of the point where this line crosses \(y = -9\) on the y-axis, that will give the root of the given equation.

Algebraic Solution

To find the root of the equation \(5x + 6 = -9\), we need to isolate x by following these steps: \Begin by subtracting 6 from both sides:\[5x+6-6 = -9 - 6 \Rightarrow 5x = -15\]Now, to isolate x, divide each side by 5: \[\frac{5x}{5} = \frac{-15}{5} \Rightarrow x = -3\]

Verification

To verify this answer, substitute \(x = -3\) into the equation and check if both sides balance. \[5*(-3) + 6 = -15 + 6 = -9\]The equality holds. Hence, x = -3 is validated which should match with the x-coordinate from the graphical solution.

Key concepts

Graphical Solution in Algebra

The graphical solution in algebra is a visual method of finding the roots of equations by plotting them on a coordinate plane. Imagine you are given the equation: \(5x + 6 = -9\). To solve this graphically, start by interpreting the left side of the equation as a function, \(y = 5x + 6\). This function represents a line, and to sketch it, you need just two points. The first point occurs when \(x = 0\), leading to \(y = 6\), so you have the point \(0, 6\). The second point is found by setting \(y = 0\) to find the \(x\)-intercept, which is \(x = -1.2\) or \(\frac{-6}{5}\), resulting in the point \(\left(-1.2, 0\right)\).When these points are plotted on a Cartesian plane, you draw a line through them. The root, or solution, to the equation is where this line crosses the \(y\)-value of \(\text{-}9\) on the graph, and is represented by the \(x\)-coordinate of that intersection. This visual approach helps you understand the behavior of the function and provides an illustrative way to find the solution to the equation.

Algebraic Solution Method

Solving equations using algebraic methods involves manipulating the equation step by step to isolate the variable. Let's explore the algebraic solution to the equation \(5x + 6 = -9\). Start by eliminating the constant term, \(6\), from both sides, simplifying to \(5x = -15\). The next step is to isolate \(x\) by dividing each side of the equation by \(5\), resulting in \(x = -3\). This process, known as solving for \(x\), provides a clear cut algebraic solution. By systematically applying these algebraic operations such as subtracting, dividing, or factoring, the variable is isolated, and the solution to the equation is found without the need for visual aids. This method is powerful and necessary, especially for equations that are not easily graphed or when an exact numerical solution is required.

Verification of Solutions

After arriving at a solution, either graphically or algebraically, it's crucial to verify its accuracy. This step checks the consistency and correctness of the solution. Consider the solution \(x = -3\) for our equation \(5x + 6 = -9\). To verify this, substitute the found value of \(x\) back into the original equation: \(5(-3) + 6\). Simplifying, you get \(\text{-}15 + 6\), which equals \(\text{-}9\), confirming that the left side equals the right side of the equation. Verification ensures that the solution satisfies the original equation, indicating that no error occurred during the problem-solving process. It's a simple yet essential step to validate the solution, whether obtained graphically or algebraically.