Question

date: 1/14/25
write a linear equation in slope - intercept form given
each point and slope.

1. (5, -2); slope = -1 \t\t\t 2. (-8, -1); slope = \\(\frac{3}{4}\\)
2. (-10, 5); slope = \\(-\frac{6}{5}\\) \t\t 4. (-7, -8); slope = \\(\frac{1}{2}\\)

Explanation:

Problem 1: \((5, -2)\); slope \(= -1\)

Step 1: Recall slope - intercept form

The slope - intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We know that \(m=-1\) and the point \((x,y)=(5, - 2)\) lies on the line. Substitute \(x = 5\), \(y=-2\) and \(m=-1\) into the equation \(y=mx + b\).
\(-2=-1\times5 + b\)

Step 2: Solve for \(b\)

Simplify the right - hand side of the equation: \(-2=-5 + b\). Add 5 to both sides of the equation to isolate \(b\).
\(b=-2 + 5=3\)

Step 3: Write the equation

Since \(m=-1\) and \(b = 3\), the equation in slope - intercept form is \(y=-x + 3\)

Problem 2: \((-8,-1)\); slope \(=\frac{3}{4}\)

Step 1: Substitute into slope - intercept form

The slope - intercept form is \(y = mx + b\). We know that \(m=\frac{3}{4}\), \(x=-8\) and \(y = - 1\). Substitute these values into the equation:
\(-1=\frac{3}{4}\times(-8)+b\)

Step 2: Solve for \(b\)

Simplify \(\frac{3}{4}\times(-8)=-6\). So the equation becomes \(-1=-6 + b\). Add 6 to both sides: \(b=-1 + 6 = 5\)

Step 3: Write the equation

Since \(m=\frac{3}{4}\) and \(b = 5\), the equation is \(y=\frac{3}{4}x+5\)

Problem 3: \((-10,5)\); slope \(=-\frac{6}{5}\)

Step 1: Substitute into slope - intercept form

Using \(y=mx + b\), with \(m =-\frac{6}{5}\), \(x=-10\) and \(y = 5\). Substitute these values:
\(5=-\frac{6}{5}\times(-10)+b\)

Step 2: Solve for \(b\)

Simplify \(-\frac{6}{5}\times(-10)=12\). The equation becomes \(5 = 12 + b\). Subtract 12 from both sides: \(b=5-12=-7\)

Step 3: Write the equation

Since \(m =-\frac{6}{5}\) and \(b=-7\), the equation is \(y=-\frac{6}{5}x-7\)

Problem 4: \((-7,-8)\); slope \(=\frac{1}{2}\)

Answer:

s:

1. \(y=-x + 3\)
2. \(y=\frac{3}{4}x+5\)
3. \(y=-\frac{6}{5}x-7\)
4. \(y=\frac{1}{2}x-\frac{9}{2}\)