For each system find the values of a and b so that the solution set to the system is {(5,12)}

step 1 So in this problem, we're told that the ordered pair 512 is the solution to the given system of

For each system find the values of a and b so that the...

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. In most contexts, systems are solved to find where the graphs of these equations intersect, which is a foundational concept in algebra and analytical geometry.

Unique Intersection or Unique Solution

When a system of linear equations has a unique solution, it means that the graphs of the equations (typically lines in two dimensions) intersect at exactly one point. This notion is important for understanding conditions under which the system of equations is consistent and determinate, and it often involves ensuring that the slopes of the lines are not equal.

Slope-Intercept Form

The slope-intercept form of a linear equation, given by y = mx + c, is a standard format that identifies the slope (m) and the y-intercept (c) of a line. Understanding this form is crucial because it directly shows how changes in the slope and intercept affect the position and orientation of the line on a graph.

Substitution Method

The substitution method involves replacing one variable in an equation with an equivalent expression obtained from another equation. This technique is widely used to reduce a system of equations to a single equation in one variable, making it simpler to identify the solution. It is particularly useful when the solution involves parameters that need to be determined.

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