91Ó°ÊÓ

When you have two or more equations working together, they form a system of equations. These equations share common variables. For instance, in the given exercise, we have two functions, \( g(x) = 3x - 5 \) and \( h(x) = \frac{7x + 10}{4} \). By solving these together, we can find the point(s) where their graphs intersect, meaning they have the same value for both \(x\) and \(y\) at those points.

To solve a system of equations:

• Look for a way to make the equations simpler.
• Combine them in a manner that eliminates one of the variables.
• Solve the simplified equation to find the value of one variable.

In this particular case, we start by setting the two equations equal to each other, as they must be equal at their intersection point.

The next step in solving our system of equations is to isolate one variable. This process of isolation helps us find the value of that variable. Here are the steps performed in the exercise:

1. **Set the equations equal**: Since \( g(x) = h(x) \) at the intersection point, we set \( 3x - 5 = \frac{7x + 10}{4} \).

2. **Eliminate the fraction**: Multiply every term by 4 to clear fractions, changing the equation to \( 4(3x - 5) = 7x + 10 \), which simplifies to \( 12x - 20 = 7x + 10 \).

3. **Isolate x**: Rearrange the terms to solve for \( x \). Subtract \( 7x \) from both sides to get \( 5x = 30 \). Finally, divide both sides by 5 to get \( x = 6 \).

Once we have the value for \( x \), we substitute it back into one of the original equations to find the corresponding value of \( y \).

Function intersection points occur where the graphs of two functions cross each other. This means they share common \( x \) and \( y \) values at these points. To find where this happens, we follow several key steps:

1. **Find the x-coordinate**: Use algebraic methods to find the \( x \) value where the two functions are equal. In the given problem, this corresponds to \( x = 6 \).

2. **Calculate corresponding y-coordinate**: Substitute the found value of \( x \) back into one of the original functions. Here, substituting \( x = 6 \) into \( g(x) = 3x - 5 \) gives us \( g(6) = 18 - 5 = 13 \).

3. **Verification**: Always check the solution by substituting the \( x \) value into the second function to ensure they produce the same \( y \) value. Substituting \( x = 6 \) into \( h(x) \), we get \( h(6) = \frac{42 + 10}{4} = 13 \), confirming our solution.

Therefore, the intersection point of the functions \( g(x) \) and \( h(x) \) is \( (6, 13) \), which helps understand how functions can intersect on a graph.