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Chapter 3: Problem 73

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

Short Answer

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

Step by step solution

01

Identify the equation

The given equation is \(6x + 3 = 0\). It's a linear equation in one variable \(x\).
02

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 \(x = -\frac{1}{2}\).
03

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.

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Key Concepts

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

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!

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Most popular questions from this chapter

Find a formula that expresses the perimeter of a square \(P\) as a function of the length of its side \(s\).

Find all intercepts for each line. Some of these lines have only one intercept. $$3 x-5 y=15$$

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=\frac{2}{3} x+1\\\ &y=-\frac{3}{2} x+3 \end{aligned} $$

Find all intercepts for each line. Some of these lines have only one intercept. $$x+2 y=100$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(0) $$
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