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Chapter 11: Problem 14

Find the \(x\) intercept for \(-6 x+5 y=30\) A. \((-5,0)\) B. \((0,6)\) c. \((0,-6)\) \(D .(5,0)\)

Short Answer

Expert verified
The x-intercept is \((-5, 0)\), so the answer is option A.

Step by step solution

01

Understanding the X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is 0. So, we will set \(y = 0\) in the equation to find the x-intercept.
02

Substitute Y with Zero

Set \(y = 0\) in the equation \(-6x + 5y = 30\). This simplifies to \(-6x + 5(0) = 30\), which further reduces to \(-6x = 30\).
03

Solve for X

To isolate \(x\), divide both sides of the equation \(-6x = 30\) by \(-6\). This results in \(x = \frac{30}{-6}\), which simplifies to \(x = -5\).
04

Identify the X-Intercept Point

Now that we know \(x = -5\) and \(y = 0\), the x-intercept point of the line is \((-5, 0)\). Compare this with the given options to identify the correct answer.

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Key Concepts

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

Linear Equations
A linear equation is simply an equation that forms a straight line when plotted on a coordinate plane. Typically, it is represented in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The fascinating aspect of linear equations is how they describe a relationship between two variables, usually \(x\) and \(y\).
  • Each coefficient, or number in front of the variable, helps determine the slope and position of the line on the graph.
  • The line itself represents all the possible solutions to the equation.
  • For solving such equations, we often look for specific points like x-intercepts and y-intercepts which help us understand the behavior of the line on the graph.
A linear equation can be easily manipulated to find these intersection points, helping solve real-world problems that involve rates or relationships between quantities.
Solving for X
When solving for \(x\) in a linear equation, the primary goal is to isolate \(x\) on one side of the equation. This process generally involves a few straightforward algebraic steps.
  • First, you look to simplify the equation by moving terms around. This could mean moving constant terms to one side and variable terms to the other.
  • Next, you simplify further by performing arithmetic operations such as addition, subtraction, multiplication, or division to get \(x\) by itself.
  • For example, if you have an equation like \(-6x + 5y = 30\) and you need to find \(x\) when \(y = 0\), you substitute \(y\) with zero and then solve for \(x\), as demonstrated in the solution steps given.
It's a methodical approach that, once understood, empowers you to solve not only simple equations but also more complex problems effectively.
Coordinate Geometry
Coordinate geometry is a vital branch of geometry where points are defined on a plane using an ordered pair of numbers, called coordinates. Understanding this concept helps connect algebra with geometric properties.
  • The x-axis and y-axis create a grid called the coordinate plane where each point is represented as \((x, y)\).
  • When finding the x-intercept of a line, you specifically look at where the line meets the x-axis (where \(y = 0\)).
  • In our example, by setting \(y = 0\) in the equation and solving for \(x\), we found the x-intercept to be \((-5, 0)\). This point tells us where the line crosses the x-axis.
These intercepts provide valuable information about where a line crosses the axes, which is essential in graphing and understanding the relationship between algebraic equations and graphical representations.

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Most popular questions from this chapter

Which point is a solution for the system? $$ \begin{array}{l} 3 x-y=10 \\ 2 x+y=-5 \end{array} $$ A. \((4,2)\) \(B,(1,3)\) cD \((1,-7)\) D. \((-5,0)\)

The slope of a horizontal line is A. 0 B. 1 C. undefined \(D,-1\)

Find \(x\) when \(y=-3\) for \(x+3 y=-10\) A. \(x=1\) B. \(x=2\) \(C . x=-1\) D. \(x=-2\)

Find \(y\) when \(x=-3\) for \(4 x+5 y=-12\) A. 5 B. 0 c. \(-2\) \(D .-3\)

Which is a solution to \(-3 x+2 y=6 ?\) A. \((2,0)\) B. \((-4,9)\) \(C .(4,9)\) \(D \cdot(0,-3)\)
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