To solve the equation [tex]\(-\frac{2}{x-1}=4\)[/tex] step-by-step, we can follow these algebraic steps:
1. Multiply both sides by [tex]\((x-1)\)[/tex] to eliminate the denominator:
[tex]\[
-\frac{2}{x-1} \cdot (x-1) = 4 \cdot (x-1)
\][/tex]
[tex]\[
-2 = 4(x-1)
\][/tex]
2. Distribute the 4 on the right side:
[tex]\[
-2 = 4x - 4
\][/tex]
3. Add 4 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-2 + 4 = 4x
\][/tex]
[tex]\[
2 = 4x
\][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{4} = x
\][/tex]
[tex]\[
x = \frac{1}{2}
\][/tex]
So, the solution to the equation [tex]\(-\frac{2}{x-1}=4\)[/tex] is [tex]\(x = 0.5\)[/tex].
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To determine which graph shows this system of equations and its solution:
1. Identify the graphs of the two functions:
- The first function derived from the equation is [tex]\(y = -\frac{2}{x-1}\)[/tex].
- The second function is [tex]\(y = 4\)[/tex].
2. Plot the functions on the graph:
- [tex]\(y = -\frac{2}{x-1}\)[/tex]: This will be a hyperbola that has a vertical asymptote at [tex]\(x = 1\)[/tex] and a horizontal asymptote at [tex]\(y = 0\)[/tex].
- [tex]\(y = 4\)[/tex]: This is a horizontal line at [tex]\(y = 4\)[/tex].
3. Find their intersection:
- The point where these two graphs intersect is the solution to the system of equations.
Based on the solution [tex]\(x = 0.5\)[/tex]:
- On the graph, you should see the hyperbola intersecting the horizontal line [tex]\(y = 4\)[/tex] at the point [tex]\((0.5, 4)\)[/tex].
Thus, the correct graph will clearly display the hyperbola [tex]\(y = -\frac{2}{x-1}\)[/tex] intersecting the line [tex]\(y = 4\)[/tex] at the point [tex]\((0.5, 4)\)[/tex], which indicates the solution [tex]\(x = 0.5\)[/tex].
