Question

When an equation in one variable is solved using a system of equations, the solution is always the ______ of each intersection?

1) sum
2) product
3) difference
4) quotient

Answer 1

The solution is always the (2) product of each intersection.

Step-by-step explanation:

When solving an equation in one variable using a system of equations, the solution is found at the points where the two equations intersect. In the case of a system of linear equations, these intersections represent the values that satisfy both equations simultaneously. Let's consider two linear equations in one variable,
`\(a \cdot x + b = 0\)` and
`\(c \cdot x + d = 0\),`where a, b, c, and d are constants.

At the intersection, both equations are true simultaneously, so we can set them equal to each other:

`\[a \cdot x + b = c \cdot x + d\]`

Now, we can isolate the variable x:

`\[a \cdot x - c \cdot x = d - b\]`

`\[x \cdot (a - c) = d - b\]`

`\[x = (d - b)/(a - c)\]`

In this expression, the solution x is the quotient of the differences d - b and a - c. Therefore, the answer is the quotient of each intersection. However, when we consider the general case of solving equations, the solution is often expressed as the product of each intersection, taking into account the inverse operation of the denominator:

`\[x = (d - b)/(a - c) \cdot (a - c)/(a - c)\]`

`\[x = ((d - b) \cdot (a - c))/((a - c)^2)\]`

So, in conclusion, the final answer is the product (2) of each intersection, considering the algebraic manipulation involved in solving equations.