Linear Functions and Graphs：練習問題

What is the slope of the straight line with equation \(2x - 5y + 10 = 0\)?

The diagram shows the straight line \(L\). Find the equation of the line \(L\).

In the figure, the straight lines \(L_1: y = m_1 x + c_1\) and \(L_2: y = m_2 x + c_2\) intersect at a point in the second quadrant. Which of the following statements must be true?

Find the \(y\)-intercept of the straight line passing through the points \((0, -4)\) and \((3, 2)\).

A straight line \(L_1\) passes through the points \((-1, 4)\) and \((3, -4)\). If a second line \(L_2\) is perpendicular to \(L_1\) and passes through the point \((2, 1)\), find the equation of \(L_2\).

The line \(L_1\) passes through \(A(1, 4)\) and \(B(k, 10)\). If \(L_1\) is parallel to the line \(L_2: 3x - y + 7 = 0\), find the value of \(k\).

The line \(L_1: y = mx + c\) is reflected across the line \(y = x\) to produce line \(L_2\). If \(L_1\) passes through \((0, 4)\) and \((2, 0)\), find the equation of \(L_2\).

Consider a straight line passing through the point \((2, -3)\) that is perpendicular to the line defined by the equation \(4x - 5y + 10 = 0\). Determine the \(y\)-intercept of this new line.

A function is defined by the linear equation \(L_1: 2x - 3y + 12 = 0\).
(a) Find the x-intercept and y-intercept of the line \(L_1\).
(b) Calculate the area of the triangle formed by the line \(L_1\) and the coordinate axes.
(c) Another line \(L_2\) passes through the origin and is parallel to \(L_1\). Write the equation of \(L_2\).

Two lines \(L_1\) and \(L_2\) are given by \(y = 2x + k\) and \(x + 2y = 10\).
(a) If the lines intersect at the point \(P(2, b)\), find the values of \(b\) and \(k\).
(b) Find the area of the triangle formed by \(L_1\), \(L_2\), and the y-axis.
(c) Determine if the line segment connecting the origin to point \(P\) is perpendicular to \(L_2\).