Systems of Equations

A set of two or more equations with the same variables is called a ______.

If a consistent system of equations has exactly one solution, it is said to be ______.

If a consistent system of equations has an infinite number of solutions, it is ______.

A system of equations is ______ if it has no ordered pair that satisfies both equations.

For the system of equations $y = -3x + 1$ and $y = x - 3$, the graph shows two lines intersecting at one point, therefore the system is consistent and ______.

For the system of equations $y = 1/2x + 2$ and $y = 1/2x - 1$, the graphs of these two lines are parallel, there is no solution of the system. Therefore, the system is ______.

A system of two linear equations can have one solution, an infinite number of solutions, or ______.

A system of equations is ______ if it has at least one ordered pair that satisfies both equations.

When two lines on a graph have the same slope but different y-intercepts, they are ______, indicating that the system of equations has no solution.

A system of equations with at least one solution is described as ______, while a system with no solution is described as inconsistent.

If two equations in a system represent the same line, the system has infinitely many solutions and is considered ______ and ______.

For the system of equations: $y = 6x + 10$ and $y = 6x + 4$, the slopes are the same, y-intercepts are different, the lines are parallel, and the system has ______.

When solving a system of equations by graphing, the solution is represented by the ______ at which the graphs of the equations intersect.

A system of equations with different slopes, different y-intercepts, and one solution is considered ______ and ______.

The equations $4y - 6x = 16$ and $9x - 6y = -24$, when both converted to slope-intercept form are the same line, therefore, there are ______ solutions.

For the system of equations: $4x - 8y = 16$ and $6x - 12y = 5$ to determine if there is a solution, the solution is ______.

The first step in solving a system of equations using the substitution method is to solve one of the equations for one of the ______.

In the equation $x + 4y = 18$, when solving for $x$, the resulting equation is $x = 18 - ______$.

When substituting $18 - 4y$ for $x$ in the equation $5x - 3y = -25$, the new equation becomes $5(18 - 4y) - 3y = ______$.

If solving a system of equations leads to the identity $-8 = -8$, the system has an ______ number of solutions.

The key concept of the elimination method using addition involves writing the system so that like terms with ______ coefficients are aligned.

In the first elimination example, the equations $3x + 5y = 11$ and $5x - 5y = 5$ are added together. The $y$ terms are ______ from the equation.

After eliminating a variable, the next step is to substitute the value from Step 2 into one of the equations and solve for the other ______.

If $x = -4$ and $y = -5$, the solution to the system of equation should be written as the coordinate $(-4, ______)$.

When graphing a system of equations, if the lines have the same slope but different y-intercepts, the system has ______.

A system of equations is considered ______ if it has at least one solution.

In the substitution method, after solving one equation for one variable, the next step is to ______ the resulting expression into the other equation.

If two lines in a system have different slopes and different y-intercepts, the system has ______ solution.

The equations $y=(3/2)x+6$ and $y=(3/2)x-2$ yield equations that have the same ______ and different y-intercepts, which means the graphs have no solution.

If the graphs of two equations in a system are the same line, then the system has a(n) ______ number of solutions.

In the context of systems of equations, the point (5, 4) is the ______ if substituting 5 for x and 4 for y into the equations makes both equations true.

If a system of equations has the same slope and same y-intercept, then it has a(n) ______ number of solutions and it is a dependent system.

The first step in solving systems of equations by elimination with subtraction is to write the system so like terms with the same ______ are aligned.

When using the elimination method with subtraction, you ______ one equation from the other to eliminate a variable.

After eliminating a variable using subtraction, you should ______ the resulting equation to find the value of the remaining variable.

Once you find the value of one variable, ______ it into one of the original equations to solve for the other variable.

The solution to a system of equations is written as an ______.

In the elimination method using multiplication, the first step involves multiplying at least one equation by a ______ to obtain opposite terms.

After multiplying equations by a constant, you ______ the equations together to eliminate one of the variables.

The goal of using elimination with either subtraction or multiplication is to ______ one variable and solve for the other.

When solving a system of equations using elimination, the goal is to ______ one of the variables.

In the elimination method, if no variable has matching or opposite coefficients, you can multiply one or both equations by a ______ to create matching or opposite coefficients.

When graphing systems of inequalities, the solution is represented by the ______ of the graphs of the inequalities.

In a system of inequalities, a ______ line on the graph indicates that the points on the line are not included in the solution.

To solve the system: $10x + 5y = 30$ and $5x - 3y = -7$, you could multiply the second equation by -2. This results in: $-10x + 6y = ______$

When solving the system of inequalities, if a test point does not satisfy an inequality, then the entire region containing that test point is ______ as part of the solution.

In solving systems of inequalities by graphing, a solid boundary line indicates the solutions on the line are ______.

To eliminate $x$ from this system: $3x + 4y = -22$ and $-2x + 3y = -8$, you could multiply the first equation by 2 and the second equation by 3. After multiplying, these equations become $6x + 8y = -44$ and $-6x + 9y = ______$

System of Equations

Solution of a System

Consistent System

Independent System

Dependent System

Inconsistent System

One Solution (Graphically)

No Solution (Graphically)

Parallel Lines (System of Equations)

Intersecting Lines (System of Equations)

Comparing Equations

Solution of a System (Graphically)

One Solution (Independent)

Infinite Solutions (Dependent)

Method of Substitution

Step 1: Substitution

Step 2: Substitution

Step 3: Substitution

Substitution Method

Steps for Substitution

Infinite Solutions (Substitution)

Elimination Method

Steps for Elimination (Addition)

Step 1 of Elimination (Addition)

Step 2 of Elimination (Addition)

Step 3 of Elimination (Addition)

Elimination

Elimination by Subtraction Steps

Alignment Before Subtraction

Elimination by Multiplication

Elimination by Multiplication

System of Inequalities

Elimination by Multiplication - Steps

Solution of a System of Inequalities

Solution to a System of Equations

Graphing Systems of Inequalities

Elimination by Addition

Dashed Boundary Line

Coefficient

Solid Boundary Line

Testing Points