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Chapter 5: Problem 75

Graph. $$6 x=3$$

Short Answer

Expert verified
x = \frac{1}{2}

Step by step solution

01

- Isolate the Variable X

To solve for the variable, we need to isolate it on one side of the equation. Start by dividing both sides by 6:\[ 6x = 3 \]Divide both sides by 6:\[ x = \frac{3}{6} \]Simplify the fraction to get \( \frac{1}{2} \).
02

- Simplify

Simplify the fraction from Step 1 to get the value of \( x \):\[ x = \frac{1}{2} \]
03

- Graph the Solution

On a number line or coordinate plane graph, mark the point where \( x = \frac{1}{2} \). If using a coordinate plane, the point would be along the x-axis at \( (0.5, 0) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving linear equations, one of our primary goals is to isolate the variable we want to solve for. This means getting the variable by itself on one side of the equation. For the equation given:\[6x = 3\]We want to isolate \(x\). To do this, we divide both sides of the equation by 6. This step ensures that the variable \(x\) is left alone on one side.After dividing by 6, we get:\[x = \frac{3}{6}\]This process simplifies our problem, making it easier to find the solution. Remember, whatever operation you perform on one side of the equation, you must also do to the other side. This keeps the equation balanced and ensures accuracy.
Fraction Simplification
Simplifying fractions is an essential part of solving equations. It makes the numbers easier to work with and simplifies our final answer. Once we have:\[x = \frac{3}{6}\]We need to simplify this fraction. To do so, we find the greatest common divisor (GCD) of the numerator (3) and the denominator (6). The GCD of 3 and 6 is 3. We divide both the numerator and the denominator by their GCD:\[\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}\]Thus, we get the simplified form. Simplifying helps ensure the solution is in the simplest form possible, making it easier to understand and interpret.
Graphing on Coordinate Plane
Graphing the solution on a coordinate plane can visualize the result. For our equation, we determined that:\[x = \frac{1}{2}\]To graph this on a coordinate plane, remember that this is a one-variable solution, and so it lies along the x-axis. Specifically, it is at the point where \(x\) is 0.5. Follow these steps to graph it:
  • Draw a horizontal line (the x-axis) and a vertical line (the y-axis) intersecting at the origin (0,0).
  • Locate 0.5 on the x-axis.
  • Mark the point at (0.5, 0).
And that's it! This visual representation helps reinforce your understanding of the solution and how it fits within the coordinate plane.

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