Power Standards Quiz 38 Examples

Aaron Bucko
11 Mar 201505:39

TLDRMr. Buckle demonstrates how to solve a system of equations by graphing, focusing on Power Standards Quiz 38. He explains that the system consists of two equations which intersect at a point, representing the solution. The first equation, y=2x-4, is graphed by plotting the y-intercept of -4 and using a slope of 2, represented as rising 2 units for every 1 unit run. The second equation, y=-3x+1, is graphed with a y-intercept of 1 and a slope of -3, depicted as descending 3 units for every 1 unit run. The intersection point of the two lines, (1, -2), is the solution to the system, which satisfies both equations.

Takeaways

  • 📈 The script provides examples of solving a system of equations by graphing for Power Standards Quiz 38.
  • 🔢 A system of equations consists of two equations, and the goal is to find their common intersection point.
  • 📌 The first equation is y = 2x - 4, which is in slope-intercept form with a y-intercept of -4 and a slope of 2.
  • 📍 The second equation is y = -3x + 1, also in slope-intercept form with a y-intercept of 1 and a slope of -3.
  • 🎨 To graph the first equation, start at the y-intercept -4 and use the slope to find additional points by rising and running.
  • 🖼️ The line drawn from the points obtained will represent the graph of the equation y = 2x - 4.
  • 📊 Similarly, graph the second equation by starting at the y-intercept 1 and using the slope to find points for the graph.
  • ✅ The solution to the system is the point where the two lines intersect, which in this case is (1, -2).
  • 📝 If the point (1, -2) is substituted into both equations, it will satisfy both, confirming it as the solution.
  • 🔄 The process is repeated with another set of equations, y = 9x - 9 and y = -5x + 5, to find their intersection at (1, 0).
  • 💡 Solving systems of equations by graphing involves visualizing their intersection points to find the solutions.

Q & A

  • What is the main topic of the video?

    -The main topic of the video is solving systems of equations by graphing, specifically for power standards quiz 38.

  • What are the two equations given in the example?

    -The two equations given are y = 2x - 4 and y = -3x + 1.

  • What does it mean when there are two equations?

    -When there are two equations, it is called a system of equations.

  • What is the goal when solving a system of equations by graphing?

    -The goal is to find the point where the two equations intersect, as this point is the solution to the system.

  • How is the slope of the first equation interpreted in the context of graphing?

    -The slope of the first equation (2) is interpreted as 'rise over run' or a fraction, and it is represented as 2/1 on the graph.

  • What is the y-intercept for the first equation and how is it used in graphing?

    -The y-intercept for the first equation is -4. It is used by plotting a point at -4 on the y-axis.

  • What is the solution to the first system of equations given in the video?

    -The solution to the first system is the point (1, -2).

  • How can you verify the solution to a system of equations?

    -You can verify the solution by plugging the coordinates of the solution point into each equation and checking if the equations hold true.

  • What are the two equations in the second example of the video?

    -The two equations in the second example are y = 9x - 9 and y = -5x + 5.

  • What is the slope and y-intercept of the second equation in the video?

    -The slope of the second equation is -5 and the y-intercept is 5.

  • What is the solution to the second system of equations demonstrated in the video?

    -The solution to the second system is the point (1, 0).

Outlines

00:00

📊 Graphing Systems of Equations - Introduction and Methodology

This paragraph introduces the concept of solving systems of equations by graphing, specifically focusing on power standards quiz 38. It explains that a system of equations involves two equations and the goal is to find the point where the graphs of these equations intersect. The explanation begins with the first equation, y = 2x - 4, which is in slope-intercept form, and details the process of plotting the y-intercept and using the slope to find additional points on the line. The concept of slope as 'rise over run' is clarified, and the process is demonstrated by finding points on the line. The second equation, y = -3x + 1, is then graphed using similar steps, emphasizing the y-intercept and the slope as 'run down'. The solution to the system is identified as the intersection point of the two lines, with the coordinates (1, -2), and the verification process by plugging the solution back into the equations is briefly mentioned. The paragraph concludes with a brief mention of another system of equations, highlighting the process of graphing and finding the solution where the lines intersect.

05:03

📈 Solving Linear Equations by Graphing - Intersection Solution

This paragraph continues the discussion on solving systems of equations by graphing, focusing on a different set of equations. It begins by graphing the first equation, y = 9x - 9, with a y-intercept of -9 and a slope of 9 (treated as 9/1). The process of finding points on the line by moving up or down based on the slope is demonstrated. The second equation, y = -5x + 5, is then graphed with a y-intercept of 5 and a slope treated as -5/1. The intersection point of these two lines is found to be (1, 0), and the solution is verified by plugging the coordinates back into the equations, resulting in true statements. The paragraph emphasizes the method of graphing as an effective way to solve systems of equations and find their solutions.

Mindmap

Keywords

💡System of Equations

A system of equations refers to a set of two or more equations that need to be solved simultaneously. In the context of the video, the system consists of two linear equations with the goal of finding the point where their graphs intersect, which represents the solution to the system. The video demonstrates this by graphing the equations y = 2x - 4 and y = -3x + 1, showing that their intersection point is (1, -2).

💡Graphing

Graphing is the process of visually representing the relationship between variables using a coordinate system. In the video, graphing is used to visually represent the equations y = 2x - 4 and y = -3x + 1 on a graph, which helps to find the solution by identifying the point where the two lines intersect.

💡Slope Intercept Form

The slope intercept form of a linear equation is a way of writing the equation where the coefficient of the independent variable (x) is written first, followed by the slope of the line, with the y-intercept (the point where the line crosses the y-axis) written as a constant term. In the video, both equations are in slope intercept form, such as y = 2x - 4, where 2 is the slope and -4 is the y-intercept.

💡Intersection Point

The intersection point is the point in the coordinate system where two lines or curves meet. In the context of the video, the intersection point is the solution to the system of equations because it represents the coordinates where the two lines intersect, satisfying both equations simultaneously.

💡Slope

The slope of a line represents the rate of change between the dependent variable (usually y) and the independent variable (usually x). It is often described as 'rise over run,' indicating how much the y-value increases for every unit increase in the x-value. In the video, the slope is used to find additional points on the line for graphing purposes.

💡Y-Intercept

The y-intercept is the point at which a line crosses the y-axis in a coordinate system. It is the value of y when x is equal to zero. In the video, the y-intercept is used as the starting point for graphing each equation and is a key component in identifying the intersection point of the system of equations.

💡Coordinate System

A coordinate system is a grid used to plot points and represent positions or locations in a two-dimensional space. It consists of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), with each axis divided into equal units. In the video, the coordinate system is used to graph the equations and find the intersection point.

💡Linear Equation

A linear equation is an algebraic equation in which the highest power of the variable is one. It represents a straight line when graphed on a coordinate system. The video focuses on solving systems of linear equations by graphing them.

💡Solution

In the context of the video, the solution refers to the values of the variables that satisfy both equations in the system simultaneously. The solution to a system of equations is typically represented as an ordered pair of numbers, with the first number being the value for the x-coordinate and the second number being the value for the y-coordinate.

💡Rise Over Run

Rise over run is a way to describe the slope of a line in terms of how much it rises vertically (rise) for every unit it runs horizontally (run). This phrase is used to calculate the slope when the change in y is divided by the change in x between two points on the line.

💡Graph

A graph is a visual representation of data or mathematical functions, typically using a coordinate system. In the context of the video, a graph is used to visually represent the equations and find their solution by showing the intersection point of the lines.

Highlights

Mr. Buckle introduces a method for solving a system of equations by graphing.

The system of equations consists of two linear equations: y = 2x - 4 and y = -3x + 1.

The goal is to find the intersection point of the two equations, which represents the solution.

The first equation is in slope-intercept form, with a slope of 2 and a y-intercept of -4.

The second equation also follows the slope-intercept form, with a slope of -3 and a y-intercept of 1.

To graph the first equation, start at the y-intercept (-4) and use the slope to find additional points.

The slope of 2 can be thought of as 'rise over run', or the fraction 2/1, to aid in graphing.

The line for y = 2x - 4 is drawn by finding points that maintain the slope of 2/1.

Similarly, the second equation is graphed with a y-intercept of 1 and a slope of -3/1.

The solution to the system is the point where the two lines intersect, which is (1, -2) for the given equations.

Verify the solution by substituting the coordinates back into each equation to obtain true statements.

Another system of equations is presented, with y = 9x - 9 and y = -5x + 5.

The y-intercept for the first equation in the new system is -9, and the slope is 9.

The second equation in the new system has a y-intercept of 5 and a slope of -5.

Graphing the second equation involves going down 5 units and right 1 unit to find points on the line.

The intersection point of the new system's equations is (1, 0), solving the system.

Substituting the solution (1, 0) into the equations confirms the correctness of the solution.

Solving systems of equations by graphing involves finding where lines intersect, representing the solution.

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