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Chapter 2: Problem 19

For each equation, find the slope. If the slope is undefined, state this. $$ 2 x-4=3 $$

Short Answer

Expert verified
The slope is undefined.

Step by step solution

01

- Isolate the variable

Start by isolating the variable term in the given equation. Given the equation: \(2x - 4 = 3\). Add 4 to both sides to isolate \(2x\).
02

- Simplify the equation

After adding 4 to both sides, the equation becomes: \(2x = 7\).
03

- Solve for \(x\)

Divide both sides of the equation by 2 to solve for \(x\). This gives: \(x = \frac{7}{2}\).
04

- Interpret the result

The result \(x = \frac{7}{2}\) represents a vertical line at \(x = \frac{7}{2}\). Vertical lines have undefined slopes.

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Key Concepts

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

isolating variables
Isolating variables is a crucial step in solving equations. It involves manipulating the equation so that the variable of interest stands alone on one side. Let鈥檚 consider our exercise equation:
\(2x - 4 = 3\).
Our goal is to isolate \(x\). A good approach is to remove any constants on the side with the variable. In this case, add 4 to both sides, yielding:
\(2x - 4 + 4 = 3 + 4\).
This simplifies to
\(2x = 7\).
Now, the variable term \(2x\) is isolated, and we're ready for the next steps.
simplifying equations
Simplifying equations makes them easier to solve. Simplifying involves combining like terms and performing basic arithmetic operations. Continuing from the isolated variable step:
\(2x = 7\).
This equation is already simple, but if it weren鈥檛, you鈥檇 look to do things like grouping similar terms or reducing fractions.
For example, in more complex equations, you might get rid of fractions or combine like terms. Once simplified, you have a clearer path to find the variable鈥檚 value. The simpler the equation, the more straightforward the solution process.
solving linear equations
Solving linear equations means finding the value of the variable that satisfies the equation. After isolating and simplifying, you should be left with a straightforward equation. For instance, with:
\(2x = 7\).
Now, we need to solve for \(x\). To do this, divide both sides by 2:
\(x = \frac{7}{2}\).
This step can involve more complex operations like factoring, but in linear equations like this one, it鈥檚 simple arithmetic. The goal is always to get \(x\) by itself with a coefficient of 1.
vertical lines
Vertical lines have a crucial property: their slope is undefined. This is because a vertical line doesn't 'run', it only 'rises'. For example, the equation \(x = \frac{7}{2}\) represents a vertical line at \(x = \frac{7}{2}\).
Vertical lines can be tricky because they don't follow the normal slope formula \( \frac{y_2 - y_1}{x_2 - x_1} \), as the denominator (difference in x-values) is zero, leading to division by zero, which is undefined.
So,
  • Horizontal change (run) = 0
  • Slope = Undefined
Recognizing this prevents mistakes when analyzing graphs or solving for slopes in equations involving vertical lines.

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