Determine whether each statement is true or false. If the...

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $A x+2 y=2$ and $2 x+B y=10$ have graphs that intersect at $(2,-2),$ then $A=-3$ and $B=3$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution to the system is the set of variable values that simultaneously satisfy all equations. Understanding these systems is crucial as it forms the basis for analyzing and solving problems where multiple conditions must be met at once.

Intersection Point

The intersection point of two graphs represents the solution to the system of linear equations. This is the point in the coordinate plane where both lines cross, indicating that the coordinates of this point satisfy both equations. Recognizing and computing the intersection is a fundamental concept in solving systems of equations.

Substitution Method

The substitution method involves replacing a variable with its known value or an expression involving other variables in order to simplify and solve a system of equations. When a solution (such as a known intersection point) is provided, this technique is used to determine unknown coefficients or parameters in the equations.

Determining Unknown Coefficients

Determining unknown coefficients in a linear equation involves substituting known values from a solution into the equation to create an equation in terms of the unknown coefficients. Solving these equations yields the specific values that ensure the equation meets the given conditions, such as passing through a specified point.

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