Intersection of Linear Equations: f(x) = 9x

Math Problem Statement

Find the point of intersection of the two linear equations: f(x) = 9x - 4 and g(x) = -3x + 2.

From the image, we see two linear equations represented on the graph:

The graph shows the intersection of these two lines. To find the point of intersection, we solve the system of equations:

f(x) &= g(x) \\ 9x - 4 &= -3x + 2 \end{aligned}$$ ### Step 1: Combine like terms $$9x + 3x = 2 + 4$$ $$12x = 6$$ ### Step 2: Solve for $$x$$ $$x = \frac{6}{12} = \frac{1}{2}$$ ### Step 3: Solve for $$y$$ Substitute $$x = \frac{1}{2}$$ into either equation. Using $$g(x) = -3x + 2$$: $$y = -3\left(\frac{1}{2}\right) + 2 = -\frac{3}{2} + 2 = \frac{1}{2}$$ Thus, the point of intersection is: $$\left(\frac{1}{2}, \frac{1}{2}\right)$$ ### Conclusion: The two lines intersect at the point $$\left(\frac{1}{2}, \frac{1}{2}\right)$$. --- Would you like further explanation or details? Here are 5 questions to expand your understanding: 1. What does the slope of a line represent in the context of these equations? 2. How would you graph these equations manually? 3. What is the significance of the intersection point geometrically? 4. How can you verify the solution algebraically and graphically? 5. How would the equations change if the lines were parallel? **Tip:** Always check your solutions by substituting the intersection point back into both equations to ensure they satisfy both.

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Linear Equations
Systems of Equations
Intersection of Lines

Solving systems of linear equations by setting equations equal to each other

Basic principles of linear systems and graphing

Grades 8-10

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