Math

Question Find the solution to the equation 1=83x-1=8-3x. Draw a graph to help solve the equation, then find the value of xx.

Studdy Solution

STEP 1

Assumptions
1. The equation given is linear.
2. The graph will be drawn in a two-dimensional coordinate system with x and y axes.
3. The solution to the equation is the x-coordinate where the graph intersects the y-axis at y = -1.

STEP 2

To graph the equation, we need to write it in the form y = mx + b, where m is the slope and b is the y-intercept.
1=83x-1 = 8 - 3x

STEP 3

Rearrange the equation to solve for y.
y=3x+8y = -3x + 8

STEP 4

Identify the slope (m) and the y-intercept (b) from the equation.
m=3,b=8m = -3, \quad b = 8

STEP 5

Plot the y-intercept (0,8) on the graph.

STEP 6

Use the slope to determine another point on the graph. Since the slope is -3, this means for every 1 unit increase in x, y decreases by 3 units.

STEP 7

Starting from the y-intercept (0,8), move 1 unit to the right (positive x direction) and 3 units down (negative y direction) to find the second point (1,5).

STEP 8

Plot the second point (1,5) on the graph.

STEP 9

Draw a straight line through the two points. This line represents the graph of the equation y = -3x + 8.

STEP 10

To find the solution to the equation, we need to determine the x-coordinate where the graph intersects the y-axis at y = -1.

STEP 11

Since we have the graph, we can visually inspect where the line crosses y = -1. Alternatively, we can solve the equation algebraically by setting y to -1.
1=3x+8-1 = -3x + 8

STEP 12

Add 3x to both sides of the equation to isolate terms with x on one side.
3x1=83x - 1 = 8

STEP 13

Add 1 to both sides of the equation to isolate the term with x.
3x=93x = 9

STEP 14

Divide both sides by 3 to solve for x.
x=93x = \frac{9}{3}

STEP 15

Calculate the value of x.
x=3x = 3
Solution: x=3x=3
Explanation: The graph of the equation y=3x+8y = -3x + 8 is a straight line with a slope of -3 and a y-intercept of 8. The solution to the equation 1=83x-1 = 8 - 3x is the x-coordinate where this line crosses the y-axis at y = -1, which is at x=3x = 3.
Check: To verify the solution, plug x=3x = 3 back into the original equation.
1=83(3)-1 = 8 - 3(3)
1=89-1 = 8 - 9
1=1-1 = -1
The solution is correct because the left-hand side equals the right-hand side when x=3x = 3.

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