Answer :
To solve the equation [tex]\(8 - 4x = 2x - 4\)[/tex] graphically, we need to consider this equation in terms of two separate linear functions and plot them on the same coordinate plane. The points where these two lines intersect will give us the solution(s) to the equation.
Here are the detailed steps:
1. Separate the equation into two functions:
- The left-hand side (LHS) of the equation: [tex]\( f_1(x) = 8 - 4x \)[/tex]
- The right-hand side (RHS) of the equation: [tex]\( f_2(x) = 2x - 4 \)[/tex]
2. Plot both functions on the coordinate plane:
- For [tex]\( f_1(x) = 8 - 4x \)[/tex]:
- This is a linear function with a slope of [tex]\(-4\)[/tex] and a y-intercept of [tex]\(8\)[/tex]. To plot this line, you can identify two points:
- When [tex]\(x = 0\)[/tex], [tex]\( y = 8 \)[/tex] (the y-intercept).
- When [tex]\(x = 1\)[/tex], [tex]\( y = 8 - 4(1) = 4\)[/tex].
- Plot the points [tex]\((0, 8)\)[/tex] and [tex]\((1, 4)\)[/tex], then draw a line through these points.
- For [tex]\( f_2(x) = 2x - 4 \)[/tex]:
- This is a linear function with a slope of [tex]\(2\)[/tex] and a y-intercept of [tex]\(-4\)[/tex]. To plot this line, you can identify two points:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -4 \)[/tex] (the y-intercept).
- When [tex]\(x = 2\)[/tex], [tex]\( y = 2(2) - 4 = 0\)[/tex].
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((2, 0)\)[/tex], then draw a line through these points.
3. Identify the intersection point:
- Once both lines are plotted on the same graph, the point of intersection represents the solution to the equation [tex]\(8 - 4x = 2x - 4\)[/tex]. This is because it is the point where the values of [tex]\(f_1(x)\)[/tex] and [tex]\(f_2(x)\)[/tex] are equal.
4. Conclusion:
- By observing the graph and determining the coordinates of the intersection point, you obtain the value of [tex]\(x\)[/tex] which satisfies the original equation.
To summarize, the graph needed to find the solution to the equation [tex]\(8 - 4x = 2x - 4\)[/tex] is the one that displays the lines [tex]\(y = 8 - 4x\)[/tex] and [tex]\(y = 2x - 4\)[/tex]. The solution to the equation is the x-coordinate of the intersection point of these two lines.
Here are the detailed steps:
1. Separate the equation into two functions:
- The left-hand side (LHS) of the equation: [tex]\( f_1(x) = 8 - 4x \)[/tex]
- The right-hand side (RHS) of the equation: [tex]\( f_2(x) = 2x - 4 \)[/tex]
2. Plot both functions on the coordinate plane:
- For [tex]\( f_1(x) = 8 - 4x \)[/tex]:
- This is a linear function with a slope of [tex]\(-4\)[/tex] and a y-intercept of [tex]\(8\)[/tex]. To plot this line, you can identify two points:
- When [tex]\(x = 0\)[/tex], [tex]\( y = 8 \)[/tex] (the y-intercept).
- When [tex]\(x = 1\)[/tex], [tex]\( y = 8 - 4(1) = 4\)[/tex].
- Plot the points [tex]\((0, 8)\)[/tex] and [tex]\((1, 4)\)[/tex], then draw a line through these points.
- For [tex]\( f_2(x) = 2x - 4 \)[/tex]:
- This is a linear function with a slope of [tex]\(2\)[/tex] and a y-intercept of [tex]\(-4\)[/tex]. To plot this line, you can identify two points:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -4 \)[/tex] (the y-intercept).
- When [tex]\(x = 2\)[/tex], [tex]\( y = 2(2) - 4 = 0\)[/tex].
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((2, 0)\)[/tex], then draw a line through these points.
3. Identify the intersection point:
- Once both lines are plotted on the same graph, the point of intersection represents the solution to the equation [tex]\(8 - 4x = 2x - 4\)[/tex]. This is because it is the point where the values of [tex]\(f_1(x)\)[/tex] and [tex]\(f_2(x)\)[/tex] are equal.
4. Conclusion:
- By observing the graph and determining the coordinates of the intersection point, you obtain the value of [tex]\(x\)[/tex] which satisfies the original equation.
To summarize, the graph needed to find the solution to the equation [tex]\(8 - 4x = 2x - 4\)[/tex] is the one that displays the lines [tex]\(y = 8 - 4x\)[/tex] and [tex]\(y = 2x - 4\)[/tex]. The solution to the equation is the x-coordinate of the intersection point of these two lines.
