Problem 30
Question
Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((5,-3)\) and is parallel to the line whose equation is \(y=2 x+1.\)
Step-by-Step Solution
Verified Answer
The equation of the line that passes through the point (5,-3) and is parallel to the line \(y=2x+1\) is \(y=2x-13\).
1Step 1: Determine the slope
Firstly, the objective is to detect the slope \(m\) of the line that the new line is parallel to. The slope of the line \(y=2x+1\) is 2. Since parallel lines have the same slope, the slope \(m\) of the line passing through (5,-3) is 2.
2Step 2: Substituting to find the y-intercept
Next, substitute the given point (5,-3) and the slope into the equation \(y=mx+c\), then solve for c to get the y-intercept. Thus, \(-3 = 2*5 + c\), which simplifies to \(-3 = 10 + c\). After simplifying we get \(c = -13\).
3Step 3: Write the equation
Finally, the equation we seek, involving slope m and y-intercept c, in slope-intercept form is \(y=mx+c\). Substituting our m and c into this equation, it becomes \(y=2x-13\).
Key Concepts
Slope of Parallel LinesSlope-Intercept FormCalculating Y-InterceptLinear Equations Algebra
Slope of Parallel Lines
Understanding the slope of parallel lines is crucial when dealing with linear equations. When two lines are parallel, they run alongside each other at the same angle, and therefore share an identical slope. In algebra, the slope is denoted as 'm' and represents the rate at which the line rises or falls as you move along the x-axis.
For example, if we have a line with the equation \(y = 2x + 1\), its slope, or 'm', is 2. This means that for every 1 unit we move to the right (positive direction along the x-axis), the line moves up 2 units. If another line is parallel to this one, its slope must also be 2, regardless of where it is on the graph. This concept is vital for writing the equation of a line parallel to another in slope-intercept form.
For example, if we have a line with the equation \(y = 2x + 1\), its slope, or 'm', is 2. This means that for every 1 unit we move to the right (positive direction along the x-axis), the line moves up 2 units. If another line is parallel to this one, its slope must also be 2, regardless of where it is on the graph. This concept is vital for writing the equation of a line parallel to another in slope-intercept form.
Slope-Intercept Form
The slope-intercept form is the most commonly used method to represent linear equations and is written as \(y = mx + c\), where 'm' represents the slope of the line and 'c' indicates the y-intercept—the point where the line crosses the y-axis. This form is particularly useful because it gives a clear picture of the line's steepness and position on the graph with just a glance.
The y-intercept is an essential part of the equation as it provides a starting point on the graph. To sketch the line quickly, one would first mark the y-intercept on the y-axis and then use the slope to find another point. From there, you draw the line through these points to represent the equation visually.
The y-intercept is an essential part of the equation as it provides a starting point on the graph. To sketch the line quickly, one would first mark the y-intercept on the y-axis and then use the slope to find another point. From there, you draw the line through these points to represent the equation visually.
Calculating Y-Intercept
To calculate the y-intercept of a line, one can use the coordinates of a point the line passes through along with the slope. The y-intercept is denoted as 'c' in the slope-intercept form of a linear equation. If the equation of a line in slope-intercept form is \(y = mx + c\), and we know the slope and a point on the line, we can substitute the values of 'x' and 'y' from the point into the equation and solve for 'c'.
For instance, with a line passing through \((5, -3)\) and a slope of 2, we plug these values into the equation to get \(-3 = 2 \times 5 + c\). Solving for 'c', we find that \(c = -13\). Therefore, the y-intercept, where our line crosses the y-axis, is at the point \((0, -13)\).
For instance, with a line passing through \((5, -3)\) and a slope of 2, we plug these values into the equation to get \(-3 = 2 \times 5 + c\). Solving for 'c', we find that \(c = -13\). Therefore, the y-intercept, where our line crosses the y-axis, is at the point \((0, -13)\).
Linear Equations Algebra
Algebra involves manipulating symbols to solve for unknowns, and one of the fundamental skills is solving linear equations. These equations describe straight lines on a coordinate plane and can appear in various forms, with one popular form being the slope-intercept form. The process includes finding the slope, which tells us how the line tilts; determining the y-intercept, where the line meets the y-axis; and often involves using known points to solve for unknown variables.
Linear equations help in modeling and solving real-world problems. When dealing with them, it is vital to understand the relationship between variables, constants, and slope. Mastery of the steps to manipulate these elements builds the foundation for more complex algebraic concepts and applications in other areas of mathematics and science.
Linear equations help in modeling and solving real-world problems. When dealing with them, it is vital to understand the relationship between variables, constants, and slope. Mastery of the steps to manipulate these elements builds the foundation for more complex algebraic concepts and applications in other areas of mathematics and science.
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