Problem 21 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$-3 x+11=2$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 3 \). It has been confirmed algebraically and graphically.

Step 1: Rearrange equation to solve for x

Begin by subtracting 11 from both sides of the equation to isolate the term with x on one side. This results in \( -3x = 2 - 11 \) which simplifies to \( -3x = -9 \). Then, divide both sides by -3 to solve for x. This gives \( x = -9 / -3 \). Therefore, \( x = 3 \).

Step 2: Check solution graphically

To check the solution graphically, plot the equation \( -3x + 11 = 2 \) on a graph. This is the same as \( -3x + 11 - 2 = 0 \) or \( -3x + 9 = 0 \). You should see that this is a straight line that intersects the x-axis at \( x = 3 \). This confirms that \( x = 3 \) is indeed a solution to the equation.

Step 3: Final verification

Substitute \( x = 3 \) back into the original equation \( -3x + 11 = 2 \). This should satisfy the equation and lead to a true statement. Substituting \( x = 3 \) gives \( -3*3 + 11 = 2 \), which simplifies to \( 2 = 2 \), confirming that \( x = 3 \) is indeed a correct solution.

Key Concepts

Graphical Solution

To understand the concept of a graphical solution, imagine plotting an equation on a coordinate plane. When you graph an equation like \(-3x + 11 = 2\), you form a line on this plane. The solution to the equation is visually represented by where this line crosses the x-axis. This is known as the x-intercept. The process of using a graph can help you verify if the algebraic solution is accurate. By graphing both sides of the equation, they should meet at the solution point, providing a visual confirmation of the answer. Check if the line intersects the x-axis at the value found algebraically. If it does, your solution is correct.

Solving Equations

Solving algebraic equations involves finding unknown variables that make the equation true. Let's break down this process with \(-3x + 11 = 2\):

• Isolate the variable: To isolate the variable \(x\), you need to get rid of any numbers or coefficients attached to it. Start by subtracting 11 from both sides to preserve equality, giving \(-3x = -9\).
• Solve by division: To solve for \(x\), divide each side by -3, the coefficient of \(x\). This results in \(x = 3\).

Now, it's crucial to verify the correctness of this value through either substitution back into the original equation or using a graphical method as an added layer of checking.

X-Intercept

The x-intercept is a key concept when solving equations graphically. It is the point where the graph of an equation crosses the x-axis. For our equation, \(-3x + 11 = 2\), when rearranged as \(-3x + 9 = 0\), finding the x-intercept means finding the value of \(x\) where the entire equation equals zero.

• The x-intercept is often represented as \(x, 0\), indicating that the y-value at this point is zero.
• To find the x-intercept manually, substitute \(0\) for \(y\) in the equation form \(-3x + 9 = 0\), solving for \(x\) will again show \(x = 3\).

This point tells us that when graphed, the equation's curve or line will touch or cross the x-axis at this coordination. It confirms the point derived algebraically that satisfies the equation, underscoring the harmony between algebraic and graphical methods.