Problem 73 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$6 x+3=0$$

Step-by-Step Solution

Answer

The \(x\)-intercept is \((-\frac{1}{2}, 0)\). No \(y\)-intercept.

Step 1: Identify the equation

The given equation is \(6x + 3 = 0\). It's a linear equation in one variable \(x\).

Step 2: Solve for x

Isolate \(x\) by first subtracting 3 from both sides: \(6x + 3 - 3 = 0 - 3\), which simplifies to \(6x = -3\).Next, divide both sides by 6:\(x = \frac{-3}{6}\), which simplifies to \(x = -\frac{1}{2}\).

Step 3: Find the intercepts

Since the equation only involves \(x\) and no \(y\) term, there is no \(y\)-intercept. The \(x\)-intercept occurs where \(x = -\frac{1}{2}\). Thus, the \(x\)-intercept is \((-\frac{1}{2}, 0)\). There are no other intercepts.

Key Concepts

headline of the respective core concept

Linear equations are fundamental in math, especially in algebra. These equations describe a line in a coordinate system and look like this: \( ax + b = 0 \), where \( a \) and \( b \) are constants.
They are considered 'linear' because they graph as straight lines.
Linear equations typically have one variable, usually \( x \). Linearity means the variable is only raised to the first power.

• Example: 6x + 3 = 0 is a linear equation.
• We can have more than one variable: ax + by = c

Understanding how to solve them is crucial for tackling more complex problems in algebra.

headline of the respective core concept

The x-intercept of a line is where it crosses the x-axis. At this point, \( y = 0 \) because we're on the horizontal axis.
To find the x-intercept, we set \( y \) to zero in the equation and solve for \( x \).
For the equation \( 6x + 3 = 0 \), since it doesn't include \( y \), everything is about \( x \).

• First, isolate x: \( 6x + 3 = 0 \)
• Next, subtract 3: \( 6x = -3 \)
• Then divide by 6: \( x = -\frac{1}{2} \)

So, for this equation, the x-intercept is \( -\frac{1}{2} \). In coordinate form, we write this intercept as \( (-\frac{1}{2}, 0) \).

headline of the respective core concept

Solving linear equations involves finding the value of \( x \) that makes the equation true. This usually means isolating \( x \) on one side of the equation.
Here's a simple method broken down:

• Start with the equation: \( 6x + 3 = 0 \)
• Get rid of constants: Subtract 3 from both sides to get \( 6x = -3 \)
• Make \( x \) stand alone: Divide both sides by 6: \( x = -\frac{1}{2} \)

Now, we've found that \( x = -\frac{1}{2} \), which is our solution. Practice this method to get better at solving any linear equation!