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Chapter 3: Problem 53

Find an equation of the line passing through the given points. $$(0,0),(3,4)$$

Short Answer

Expert verified
The equation of the line that passes through the points (0,0) and (3,4) is \(y = \frac{4}{3}x\).

Step by step solution

01

Determine the Slope

The slope of a line passing through the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\] Substituting \((0,0)\) for \((x_1,y_1)\) and \((3,4)\) for \((x_2,y_2)\), we get \[m = \frac{4 - 0}{3 - 0} = \frac{4}{3}\]. So, the slope of the line is \(\frac{4}{3}\)
02

Use the Point-Slope Form

The point-slope form of a line's equation is \[y - y_1 = m(x - x_1)\] where \(m\) is the slope and \(x_1,y_1\) is a point on the line. Here we can use the point (0,0) so the equation simplifies to \[y = m x\]. Substituting \(\frac{4}{3}\) for \(m\) we get \[y = \frac{4}{3}x\] as the equation of the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
Calculating the slope is a fundamental step in finding the equation of a line that passes through two given points. The slope tells us how steep the line is and in which direction it slants. We use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope \( m \). Here, \((x_1, y_1)\) are the coordinates of the first point, and \((x_2, y_2)\) are the coordinates of the second point.
For the points \((0,0)\) and \((3,4)\), the calculation becomes:
  • Subtract the \( y \)-coordinates: \( 4 - 0 = 4 \).
  • Subtract the \( x \)-coordinates: \( 3 - 0 = 3 \).
  • Divide the difference in \( y \) by the difference in \( x \): \( \frac{4}{3} \).
Thus, the slope of the line is \( \frac{4}{3} \), indicating that for every 3 units we move horizontally, we move 4 units vertically.
Point-Slope Form
Once we have the slope, the point-slope form helps us write the equation for the line. It's a very useful form, especially when we have one point and the slope. The point-slope form equation is expressed as:
  • \( y - y_1 = m(x - x_1) \)
  • \((x_1, y_1)\) is a point on the line, \( m \) is the slope.
In this case, by using the point \((0,0)\), a special simplification occurs. The equation reduces to \( y = mx \) because \( y_1 \) and \( x_1 \) are zero. This means our line's equation becomes \( y = \frac{4}{3}x \), using the calculated slope \( \frac{4}{3} \). This tells us how \( y \) changes concerning \( x \). Each input for \( x \) is multiplied by \( \frac{4}{3} \) to give the corresponding \( y \).
Linear Equations
Linear equations represent straight lines in algebra. In a general context, their standard form is \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. For the example we have, the line passes through the origin \((0, 0)\), so \( b = 0 \). Therefore, our equation simplifies to \( y = \frac{4}{3}x \).
  • This specific equation does not have a constant term (the \( b \)), so the line crosses the y-axis at the origin.
  • A linear equation like this means that any change in \( x \) directly affects \( y \) with the same linear relationship and no added constants.
Such equations are valuable in modeling relationships in real-world situations where relationships are proportional, ensuring footing in both mathematical theory and practical application.

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Most popular questions from this chapter

Find an equation of the line passing through the given points. $$(1,4),(3,4)$$

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 4 \end{array}\right]$$

Use Cramer's Rule to solve the system of linear equations for \(x\) and \(y\) \\[ \begin{array}{r} k x+(1-k) y=1 \\ (1-k) x+\quad k y=3 \end{array} \\] For what value(s) of \(k\) will the system be inconsistent?

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rrr} -3 & -5 & -7 \\ 2 & 4 & 3 \\ 0 & 1 & -1 \end{array}\right]$$

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{ll} 2 & 1 \\ 3 & 0 \end{array}\right]$$
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