Problem 28 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,0)\) and \((0,2)\)

Step-by-Step Solution

Answer

The point-slope form is \(y - 0 = 1(x - (-2))\), which simplifies to \(y = x + 2\). The slope-intercept form is also \(y = x + 2\). Hence the line that passes through the points \((-2,0)\) and \((0,2)\) is described by the equation \(y = x + 2\).

Step 1: Calculation of Slope

Use the given points and the slope formula to find the slope, \(m\). Here, let's use the points, \((-2,0)\) and \((0,2)\). Then \(m = (2 - 0) / (0 - (-2)) = 2/2 = 1\)

Step 2: Write the Point-Slope Form

Substitute the slope value and the coordinates of one of the points into the point slope form of the equation, which is \(y - y1 = m(x - x1)\) to obtain \(y - 0 =1(x - (-2))\), which simplifies to \(y = x + 2\)

Step 3: Write the Slope-Intercept Form

The slope-intercept form of the equation can be obtained from the point-slope form by simplifying the equation. In this case, the slope-intercept form is already obtained \(y = x + 2\)

Step 4: Check the Answer

Ensure that the equation is correct by testing if the second point \((0,2)\) satisfies the equation. Substituting \(x = 0\), the equation turns to \(y = 0 + 2\), which gives \(y = 2\). Since this matches the \(y\) coordinate of the second point, the equation is correct.