Linear equations in 2 variables：練習題

In the \(xy\)-plane, the graph of the linear equation \(y = \frac{2}{3}x + k\) passes through the point \((6, 5)\). What is the value of the constant \(k\)?

What is the equation of the line in the \(xy\)-plane that passes through the point \((2, 5)\) and has a slope of 3?

Line \(L\) in the \(xy\)-plane is perpendicular to the line with the equation \(y = 2x + 3\). If line \(L\) passes through the point \((4, 1)\), what is the \(x\)-intercept of line \(L\)?

What is the \(x\)-intercept of the line given by the equation \(3x + 2y = 12\) in the \(xy\)-plane?

Line \(L\) passes through the points \((2, 3)\) and \((4, 7)\). What is the \(x\)-intercept of line \(L\)?

A line in the \(xy\)-plane has the equation \(2x - y = 6\). What is the value of the \(y\)-coordinate when \(x = 4\)?

In the \(xy\)-plane, line \(K\) is represented by the equation \(Ax + By = 12\). If line \(K\) passes through the points \((2, 2)\) and \((4, 0)\), what is the value of \(A - B\)?

The linear equation \( y = mx + b \) passes through the point \( (2, -3) \) and is perpendicular to the line with the equation \( 4x - 2y = 8 \). What is the value of \( b \)?

A community garden rents out rectangular plots of land. The relationship between the total monthly cost \(y\), in dollars, for a plot and the area of the plot \(x\), in square meters, is modeled by the linear equation \(y = 3x + 25\).

Part a: What is the total monthly cost, in dollars, for a plot with an area of \(40\) square meters?

Part b: According to the model, what is the fixed base fee, in dollars, charged for any plot?

In the \( xy \)-plane, line \( k \) passes through the origin and has a slope of \( \frac{2}{3} \). Line \( p \) passes through the point \( (0, 10) \) and is perpendicular to the line with the equation \( y = 2x - 5 \).

Part a: Write the equations for line \( k \) and line \( p \) in slope-intercept form.
Part b: Find the coordinates of the point \( (x, y) \) where line \( k \) and line \( p \) intersect.