Problem 29
Question
Write the slope-intercept equation of the line that passes through the given point and that is parallel to the given line. $$ (1,-2), y=-x / 2 $$
Step-by-Step Solution
Verified Answer
The equation is \( y = -\frac{1}{2}x - \frac{3}{2} \).
1Step 1: Identify the slope of the given line
The equation of the given line is in slope-intercept form: \( y = -\frac{1}{2}x \). The slope of this line is \( m = -\frac{1}{2} \). Lines that are parallel have the same slope.
2Step 2: Use the point-slope form
To find the equation of the line that is parallel and passes through the point \((1, -2)\), use the point-slope form of a line: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point and \( m \) is the slope. Substitute \( (x_1, y_1) = (1, -2) \) and \( m = -\frac{1}{2} \): \( y - (-2) = -\frac{1}{2}(x - 1) \).
3Step 3: Simplify the equation
Simplify the equation from Step 2: \( y + 2 = -\frac{1}{2}x + \frac{1}{2} \). Subtract 2 from both sides: \( y = -\frac{1}{2}x + \frac{1}{2} - 2 \). Simplify the right-hand side to get \( y = -\frac{1}{2}x - \frac{3}{2} \).
Key Concepts
Parallel LinesPoint-Slope FormLinear Equations
Parallel Lines
Parallel lines are a fascinating aspect of geometry and algebra, often making appearances in problems that involve the slope-intercept form of linear equations. The main characteristic of parallel lines in a coordinate plane is that they never intersect, which occurs because they have the same slope.
For example, if a line has a slope of \( m = \frac{1}{2} \), any line parallel to it will also have the same slope, \( m = \frac{1}{2} \). This is crucial in determining equations of lines that must be parallel to a given line. In the exercise example, when we know that the line has an equation \( y = -\frac{1}{2}x \), any parallel line we need to find will also have a slope of \( -\frac{1}{2} \).
Understanding parallel lines helps solve a variety of geometric problems and allows for the application of algebraic methods to geometry. When working through exercises like these, remember:
For example, if a line has a slope of \( m = \frac{1}{2} \), any line parallel to it will also have the same slope, \( m = \frac{1}{2} \). This is crucial in determining equations of lines that must be parallel to a given line. In the exercise example, when we know that the line has an equation \( y = -\frac{1}{2}x \), any parallel line we need to find will also have a slope of \( -\frac{1}{2} \).
Understanding parallel lines helps solve a variety of geometric problems and allows for the application of algebraic methods to geometry. When working through exercises like these, remember:
- Parallel lines have identical slopes.
- They are used to determine the equations of new lines in many algebra problems.
- Key in transformations and ability to visualize lines on a graph.
Point-Slope Form
The point-slope form is a useful tool for finding the equation of a line when you know a point on the line and the slope. Often, this is preferred when the slope and a specific point are given, as it allows for a direct substitution.
The general formula for point-slope form is: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. In our exercise, we utilized this form to find the equation of the line parallel to \(y = -\frac{1}{2}x\) and passing through the point \((1, -2)\).
Substituting the slope \(m = -\frac{1}{2}\) and the point \((1, -2)\) into the formula, you get:\[ y - (-2) = -\frac{1}{2}(x - 1) \] This equation can then be simplified to uncover the slope-intercept form of the line.
Using the point-slope form confidently:
The general formula for point-slope form is: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. In our exercise, we utilized this form to find the equation of the line parallel to \(y = -\frac{1}{2}x\) and passing through the point \((1, -2)\).
Substituting the slope \(m = -\frac{1}{2}\) and the point \((1, -2)\) into the formula, you get:\[ y - (-2) = -\frac{1}{2}(x - 1) \] This equation can then be simplified to uncover the slope-intercept form of the line.
Using the point-slope form confidently:
- Simplifies creating equations of new lines.
- A direct plug-in method, reducing error.
- Transitions easily from given data to linear equations.
Linear Equations
Linear equations form the foundation of algebra, describing the relationship between variables with a constant slope. When we talk about a linear equation in two variables, it's typically represented as \(y = mx + b\), known as the slope-intercept form.
The slope-intercept form is beneficial for quickly identifying key properties like the slope \(m\) and the y-intercept \(b\), the latter being where the line crosses the y-axis. Whether you're graphing a basic line or solving more complex systems, understanding how to work with linear equations is essential.
In our exercise, we worked with these concepts to transition an equation from point-slope form to the slope-intercept form.
Benefits of mastering linear equations:
The slope-intercept form is beneficial for quickly identifying key properties like the slope \(m\) and the y-intercept \(b\), the latter being where the line crosses the y-axis. Whether you're graphing a basic line or solving more complex systems, understanding how to work with linear equations is essential.
In our exercise, we worked with these concepts to transition an equation from point-slope form to the slope-intercept form.
Benefits of mastering linear equations:
- Essential for solving and simplifying algebraic expressions.
- Key for graphically representing data and relationships.
- Forms a basis for understanding more complex mathematical concepts like calculus.
- Makes analyzing trends easier, especially in statistics and economics.
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