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Magazine Chapter 1: Problem 27
Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(1 ; \quad y\) -intercept \(-3\)
Short Answer
Expert verified
The equation of the line is \(y = 3x - 3\).
Step by step solution
01
Understand the Intercepts
The given x-intercept is 1. This means the line passes through the point \((1, 0)\). The y-intercept is -3, so the line passes through the point \((0, -3)\).
02
Calculate the Slope
The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \((x_1, y_1) = (1, 0)\) and \((x_2, y_2) = (0, -3)\) into the formula to find the slope:\[m = \frac{-3 - 0}{0 - 1} = \frac{-3}{-1} = 3\].
03
Use Point-Slope Form
Now that we have the slope \(m = 3\), we can use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\). Using point \((1, 0)\), the equation becomes:\[y - 0 = 3(x - 1)\].
04
Simplify the Equation
Simplify the equation from the point-slope form. Start with \[y = 3(x - 1)\]. Then distribute the 3: \[y = 3x - 3\]. This is the equation of the line in slope-intercept form \(y = mx + b\).
05
Verify by Checking Intercepts
Lastly, check the x-intercept by setting \(y = 0\) and solving for \(x\) in the equation \(3x - 3 = 0\). Solving gives \(x = 1\). Check the y-intercept by setting \(x = 0\) which gives \(y = -3\). Both intercepts match, confirming the solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the X-Intercept
The **x-intercept** of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. In simpler terms, it's where the graph touches or crosses the horizon of the x-axis. This point is vital because it helps in sketching the graph of the equation or understanding properties of the line. To find the x-intercept in an equation, set the y-value to zero and solve for x. For example, if given an equation like \( y = 3x - 3 \), set \( y = 0 \) to find:
- \( 0 = 3x - 3 \)
- Solving the equation, \( 3x = 3 \)
- \( x = 1 \)
Understanding the Y-Intercept
The **y-intercept** of a line is the point where the line crosses the y-axis. At this location, the x-coordinate is always zero. This point tells us the position of the line relative to the vertical axis and is a key starting point for graphing.To determine the y-intercept, you'd set the value of x to zero in the line's equation and solve for y. For instance, consider the equation \( y = 3x - 3 \). When we set \( x = 0 \), it becomes:
- \( y = 3(0) - 3 \)
- \( y = -3 \)
Slope-Intercept Form
The **slope-intercept form** of a line's equation is expressed as \( y = mx + b \). This format is exceptionally useful for quickly identifying the slope and y-intercept of the line:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.
- The slope \( m \) is 3.
- The y-intercept \( b \) is -3.
Point-Slope Form
The **point-slope form** of a line is another useful way to express the equation of a line, especially when you know the slope and a point on the line. The formula is written as \( y - y_1 = m(x - x_1) \):
- \( m \) is the slope.
- \( (x_1, y_1) \) is a known point on the line.
- \( y - 0 = 3(x - 1) \)
- \( y = 3x - 3 \)
