Question

functions and lines
solving a linear equation by graphing
here is an equation.
\\( 0 = 4 + 2x \\)
draw a graph to help solve the equation. then find the solution to the equation.
solution: \\( x = \square \\)

Explanation:

Step1: Rewrite the equation as a function

We can rewrite the equation \(0 = 4 + 2x\) as \(y = 2x + 4\). This is a linear function in the form \(y = mx + b\), where \(m = 2\) (slope) and \(b = 4\) (y - intercept).

Step2: Find the y - intercept

The y - intercept occurs when \(x = 0\). Substituting \(x = 0\) into \(y = 2x+4\), we get \(y=2(0) + 4=4\). So one point on the line is \((0,4)\).

Step3: Find another point using the slope

The slope \(m = 2=\frac{2}{1}\), which means for every 1 unit we move to the right (increase in \(x\) by 1), we move up 2 units (increase in \(y\) by 2). Starting from \((0,4)\), if we move \(x = 1\), then \(y=4 + 2=6\), so the point \((1,6)\) is on the line. We can also use the slope to find a point to the left. If \(x=- 1\), then \(y=2(-1)+4 = 2\), so the point \((-1,2)\) is on the line.

Step4: Graph the line

Plot the points \((0,4)\), \((1,6)\), \((-1,2)\) (or other points found) and draw a straight line through them. The solution to the equation \(0 = 4+2x\) is the value of \(x\) when \(y = 0\) (since we want to find when \(4 + 2x=0\)). To find this algebraically (we can also see it from the graph), we solve \(0=4 + 2x\) for \(x\).
Subtract 4 from both sides: \(0-4=4 + 2x-4\), which gives \(-4 = 2x\).
Divide both sides by 2: \(\frac{-4}{2}=\frac{2x}{2}\), so \(x=-2\).

Answer:

\(x=-2\)