Master this deck with 20 terms through effective study methods.
Generated from uploaded pdf
The Cartesian Coordinate Plane is a two-dimensional plane formed by the intersection of a horizontal number line (x-axis) and a vertical number line (y-axis). It is named after René Descartes, a French philosopher and mathematician.
To graph a linear equation in two variables, rearrange the equation into slope-intercept form (y = mx + b), identify the slope (m) and y-intercept (b), plot the y-intercept on the y-axis, and use the slope to find another point. Draw a line through the points.
The slope in a linear equation represents the rate of change of y with respect to x. It indicates how steep the line is and the direction it goes; a positive slope means the line rises, while a negative slope means it falls.
The standard form of a linear equation is expressed as ax + by = c, where a, b, and c are integers, and a and b are not both zero. This form is useful for quickly identifying intercepts and for solving systems of equations.
To find the y-intercept of a linear equation, set x to 0 in the equation and solve for y. The resulting value of y is the y-intercept, which is the point where the line crosses the y-axis.
Two lines are parallel if they have the same slope but different y-intercepts. This means they will never intersect and maintain a constant distance apart.
To solve a system of linear equations graphically, graph each equation on the same coordinate plane. The solution to the system is the point where the two lines intersect, representing the values of x and y that satisfy both equations.
The slope-intercept form of a linear equation is expressed as y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful for quickly graphing linear equations.
To determine if a point (x, y) lies on the line represented by a linear equation, substitute the x-value into the equation and see if the resulting y-value matches the y-coordinate of the point.
The coefficients in the standard form ax + by = c relate to the graph of the line by determining its slope and intercepts. The slope can be found by rearranging the equation into slope-intercept form, and the intercepts can be found by setting x or y to zero.
The x-axis is the horizontal line that represents the set of all possible x-values, while the y-axis is the vertical line that represents the set of all possible y-values. Together, they create a framework for plotting points and graphing equations.
To convert a linear equation from slope-intercept form (y = mx + b) to standard form (ax + by = c), rearrange the equation by moving the mx term to the left side and adjusting the equation so that a, b, and c are integers.
To find the equation of a line given a point (x1, y1) and a slope m, use the point-slope form of the equation: y - y1 = m(x - x1). You can then rearrange it to slope-intercept or standard form as needed.
Two lines are perpendicular if the product of their slopes is -1. This means that one line rises while the other falls, creating a right angle at their intersection.
To identify the x-intercept of a linear equation, set y to 0 in the equation and solve for x. The resulting value of x is the x-intercept, which is the point where the line crosses the x-axis.
Graphing linear equations is important in real-world applications as it allows for visual representation of relationships between variables, making it easier to analyze trends, make predictions, and solve problems in fields such as economics, physics, and engineering.
To rearrange a linear equation to solve for y, isolate y on one side of the equation by moving all other terms to the opposite side. This often involves adding or subtracting terms and then dividing by the coefficient of y.
To find the slope of a line given two points (x1, y1) and (x2, y2), use the formula m = (y2 - y1) / (x2 - x1). This calculates the change in y over the change in x between the two points.
The y-intercept is the point where the line crosses the y-axis, representing the value of y when x is zero. It provides a starting point for graphing the line and is crucial for understanding the behavior of the equation.
The term 'linear' refers to the fact that the graph of the equation forms a straight line. Linear equations represent relationships with a constant rate of change, meaning that for every unit increase in x, there is a consistent change in y.