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Chapter 0: Problem 128

Let \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\) be two nonparallel lines \(\left(m_{1} \neq m_{2}\right) .\) What is the \(x\) -coordinate of the point where they intersect?

Short Answer

Expert verified
The x-coordinate of the intersection is \(x = \frac{b_2 - b_1}{m_1 - m_2}\).

Step by step solution

01

Set Equations Equal

To find the intersection of two lines, we need to determine where they have the same values for both their x and y coordinates. Set the equations equal to each other: \(m_1 x + b_1 = m_2 x + b_2\).
02

Rearrange to Solve for x

Rearrange the equation to isolate terms involving \(x\) on one side. We subtract \(m_2 x\) from both sides to get \((m_1 - m_2)x = b_2 - b_1\).
03

Solve for x

Divide both sides by \(m_1 - m_2\) to solve for \(x\). This gives \(x = \frac{b_2 - b_1}{m_1 - m_2}\).

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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 mathematical statements that describe a straight line when plotted on a coordinate plane. They are usually expressed in the form of \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept.
The slope \( m \) indicates how steep the line is, as well as its direction. A positive slope means the line ascends from left to right, while a negative slope indicates it descends. When the slope is zero, the line is horizontal.
The y-intercept \( b \) is the point where the line crosses the y-axis. In the equation \( y = mx + b \), as \( x = 0 \), \( y = b \). This point is an important reference to graph the line easily.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system. It provides a connection between algebra and geometry through graphs of equations and geometric shapes.
In coordinate geometry, we use a 2D plane (the Cartesian plane), consisting of an x-axis and a y-axis. Points on this plane are expressed as ordered pairs \((x, y)\). This approach helps in accurately describing spatial relationships and solving geometric problems.
  • Equations such as \( y = mx + b \) describe lines in this plane.
  • By using coordinates, we can find distances, angles, and other quantities analytically.
  • It also helps in determining intersections and dimensions of shapes formed by lines and curves.
Understanding coordinate geometry is crucial for solving problems involving geometric shapes and relationships in a structured algebraic manner.
Solving Equations
Solving equations involves finding the unknown values that satisfy given mathematical expressions. In the case of linear equations, we aim to find the value of the variable \( x \) that makes the equation true.
To solve systems of linear equations, such as finding the intersection of two lines \( y_1 = m_1x + b_1 \) and \( y_2 = m_2x + b_2 \), we set them equal to each other. This helps find the common solution for \( x \), where the lines intersect.
  • Set the equations equal: \( m_1 x + b_1 = m_2 x + b_2 \).
  • Rearrange the terms to get \( (m_1 - m_2)x = b_2 - b_1 \).
  • Isolate \( x \) and solve: \( x = \frac{b_2 - b_1}{m_1 - m_2} \).
Understanding solving equations is essential in algebra as it provides the foundation for analyzing and predicting outcomes in various mathematical problems and real-life situations.

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