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Chapter 2: Problem 45

Find the point at which the line \(f(x)=-2 x-1\) intersects the line \(g(x)=-x\)

Short Answer

Expert verified
The lines intersect at the point (-1, 1).

Step by step solution

01

Set the Equations Equal

To find the intersection of the functions, set the equations for the lines equal to each other: \[-2x - 1 = -x\]
02

Isolate the Variable

To solve for \(x\), first add \(2x\) to both sides to get:\[-1 = x\]
03

Solve for x

We already isolated \(x\) and found that \(-1 = x\). Therefore, \(x = -1\).
04

Substitute x into One of the Equations

Substitute \(x = -1\) back into either line equation, for instance, \(g(x) = -x\):\[g(-1) = -(-1) = 1\]
05

Write the Intersection Point

The intersection point occurs at \(x = -1\) and \(g(x) = 1\). Therefore, the intersection point is \((-1, 1)\).

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Key Concepts

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

Linear Equations
Linear equations are like straight paths on a coordinate plane. Each equation represents a line through points that follow a constant rate of change, known as the slope. The basic form of a linear equation is given as \( y = mx + b \), where:
  • \(m\) is the slope of the line, showing how steep the line is.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
In our exercise, the equations are \(f(x) = -2x - 1\) and \(g(x) = -x\). Here, \(f(x)\) has a slope of \(-2\), making it steeper compared to \(g(x)\), which has a slope of \(-1\). Understanding these properties helps you visualize how the lines are oriented and where they might intersect.
Solving Systems of Equations
Finding where two lines intersect involves solving a system of equations. This step is about determining a single point, if any, where both line equations hold true. To solve systems of equations like in the given exercise, you can follow a straightforward approach:
  • Set the equations equal to each other: This means combining the equations as shown: \(-2x - 1 = -x\).
  • Solve for \(x\): Simplifying gives \(-1 = x\), which means \(x = -1\).
  • Substitute back to find \(y\): Plugging \(x = -1\) into either original equation reveals \(g(x) = 1\).
By setting the equations equal, you essentially find the value of \(x\) that makes both equations valid. This process results in the intersection point where the solutions to both equations coincide.
Coordinate Geometry
Coordinate geometry, or analytic geometry, merges algebra and geometry to solve problems involving distances and shapes on a plane. With this mathematics, you can find where lines intersect, like in our exercise, by using a coordinate plane represented by axes.In our context:
  • X-axis (horizontal) and Y-axis (vertical): These allow plotting of points based on their \(x\) and \(y\) values.
  • Intersection point: It occurs where both line equations share a common \(x, y\) pair. Here, it is \((-1, 1)\).
By using coordinate geometry, you don鈥檛 just find the lines鈥� paths but also the exact point they meet. This application helps turn abstract algebraic solutions into visual, understandable graph representations.

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Most popular questions from this chapter

A farmer finds there is a linear relationship between the number of bean stalks, \(n,\) she plants and the yield, \(y\), each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form \(y=m n+b\) that gives the yield when \(n\) stalks are planted.

Write an equation for a line perpendicular to \(h(t)=-2 t+4\) and passing through the point (-4,-1)

Solve each the equation. $$ |4 x+2|=15 $$

Find the horizontal and vertical intercepts of each equation. $$ h(x)=3 x-5 $$

Solve each inequality. $$ |x-3|<7 $$
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