SAT Practice Question: A system of three equations and their graphs are shown in the xy plane below. How many solutions does the system have? Source: collegereadiness.collegeboard.org
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Solving Systems of Equations
When we solve an equation, we are looking for values that, when substituted in for the variable, make the equation true. For example, \(x=-11\) is a solution to the equation \(2x+7=-15\) because a true statement is made when \(-11\) is substituted in for \(x\). In an earlier target, we looked at equations that had no solution and infinite solutions as well.
A system of equations is a set of two or more equations. When we are looking to solve a system of equations, we want to find solutions (a value, an ordered pair, an ordered triple, etc.) that are a solution to each of the equations in the system. There are many possibilities that can happen.
This first system which consists of a linear equation and a quadratic equation has two solutions (the two points of intersection). However, if we were to shift either equation, then there may be only one or no solutions.
When we solve an equation, we are looking for values that, when substituted in for the variable, make the equation true. For example, \(x=-11\) is a solution to the equation \(2x+7=-15\) because a true statement is made when \(-11\) is substituted in for \(x\). In an earlier target, we looked at equations that had no solution and infinite solutions as well.
A system of equations is a set of two or more equations. When we are looking to solve a system of equations, we want to find solutions (a value, an ordered pair, an ordered triple, etc.) that are a solution to each of the equations in the system. There are many possibilities that can happen.
This first system which consists of a linear equation and a quadratic equation has two solutions (the two points of intersection). However, if we were to shift either equation, then there may be only one or no solutions.
The second system which consists of two linear equations has no solution because there are no common solutions (or points of intersection).
This third system which consists of a linear equation and a trigonometric equation (which you will learn about second semester) has an infinite number of solutions. Note that it is not all solutions since there are many ordered pairs that are a solution to one equation but not the other and also ordered pairs that are not solutions to either equation. However, there are an infinite number of intersection points.
In this target, we will focus on solving systems of two and three linear equations algebraically.
Solve Systems of Two Linear Equations
Solve Systems of Three Linear Equations