Problem 27 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$-\frac{3}{4} x+3=\frac{9}{4}$$

Step-by-Step Solution

Answer

By solving the given equation algebraically, we get \(x = 1\). And graphically, this solution is verified as the lines intersect where \(x = 1\).

Step 1: Simplify the Equation

First start by subtracting \(3\) from both sides of the equation so that it forms like \(ax = b\). So it becomes \(-\frac{3}{4} x + 3-3 = \frac{9}{4}-3\), which simplifies to \(-\frac{3}{4}x = -\frac{3}{4}\).

Step 2: Solve for x

Now, to solve for \(x\), divide both sides of the equation by \(-\frac{3}{4}\). This will isolate \(x\) on one side of the equation. Doing so gives \(x = (-\frac{3}{4}) / (-\frac{3}{4}) = 1\). Therefore, \(x = 1\).

Step 3: Check Solution Graphically

Plot the line for the initial equation \(-\frac{3}{4}x + 3\). Then plot a vertical line at \(x = 1\). Observe where these two lines intersect. If the intersection point is at the same value on the y-axis as when \(x = 1\) is substituted in the equation, then the solved value is correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the variable you are solving for. This enables a clear path to the solution. In the given exercise, step one involves simplifying the equation by subtracting 3 from both sides. By maintaining balance through identical operations on both sides, you achieve a simpler form. This results in:

• -\frac{3}{4}x = -\frac{3}{4}

The final step in solving the equation algebraically involves isolating the variable \(x\). We achieve this by dividing both sides by \(-\frac{3}{4}\), leading to \(x = 1\). Simplification, identifying like terms, and performing inverse operations are core elements of algebraic manipulation, making it an essential tool for solving equations.

Graphical Solution

A graphical solution provides a visual method to verify the results obtained algebraically. For this exercise, you'll plot the linear equation \(-\frac{3}{4}x + 3\) on a graph. When graphing, it's beneficial to:

• Identify the y-intercept, which is the starting point, here \(y = 3\).
• Use the slope, \(-\frac{3}{4}\), to identify another point on the line by moving 3 units down and 4 units right from the y-intercept.

This graph provides a visual confirmation. Add a vertical line at \(x = 1\) to check the intersection point. The intersection confirms our solution: where the line \(x = 1\) meets the equation's plot, confirming \(y\) is at the expected value from substitution.

Checking Solutions

Ensuring solutions are correct is crucial. This involves substituting the calculated variable back into the original equation to check if it satisfies the equation. By substituting \(x = 1\) back into the equation \(-\frac{3}{4}(1) + 3\), check if it equals \(\frac{9}{4}\).

• Perform the multiplication: \(-\frac{3}{4} \times 1 = -\frac{3}{4}\).
• Add 3: \(-\frac{3}{4} + 3 = \frac{9}{4}\).

Since both sides are equal, \(x = 1\) is confirmed as the correct solution. By both algebraically and graphically checking, you can build confidence in the validity of the solution.

Step-by-step Problem Solving

Working through problems step-by-step involves breaking down the task into manageable parts for clarity and precision. This approach allows focus on each reduction and transformation. Follow these steps:

• Simplify the equation to reduce complexity, like subtracting equal terms from both sides.
• Solve for the variable, using inverse operations to isolate it.
• Graph the equation to visualize the intersection with solutions.
• Verify by substituting the solution back into the original equation.

Each step builds on the previous one, fostering a clearer understanding of the process involved in solving linear equations. This structured approach reduces errors and enhances comprehension, making the process more intuitive.