What is a linear equation in two variables?

A linear equation in two variables is an equation that can be expressed in the form Ax + By + C = 0, where A, B, and C are constants, and A and B are not both zero.

What is the slope-intercept form of a linear equation?

The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept.

How do you find the slope from the equation 4x - 2y = 8?

Rearranging the equation to slope-intercept form gives 2y = 4x - 8, or y = 2x - 4. The slope is 2.

What is the standard form of the linear equation y = -3x + 5?

The standard form is 3x + y - 5 = 0.

What indicates that a system of equations has no solution?

If the two lines represented by the equations are parallel, they will never intersect, indicating no solution.

Solve the system: 2x + 3y = 6 and x - y = 1. 

Hint: Try substitution or elimination.

From x - y = 1, we find x = y + 1. Substituting into the first equation: 2(y + 1) + 3y = 6. This simplifies to 5y + 2 = 6, thus y = 4/5. Then, x = 1 + 4/5 = 9/5. The solution is (9/5, 4/5).

What is the graphical representation of a linear equation in two variables?

The graphical representation of a linear equation in two variables is a straight line.

Which method involves graphing the equations to find the solution?

The graphical method involves plotting the equations on a graph to find the point of intersection, which represents the solution.

What are the possible types of solutions for a system of linear equations?

The system can have one unique solution, no solution, or infinitely many solutions.

If the equations of two lines are identical, what type of solution does it imply?

If the equations are identical, it implies there are infinitely many solutions.

What is the elimination method used for?

The elimination method is used to eliminate one variable by adding or subtracting the equations to solve for the other variable.

Find the solution for the system: 3x + 4y = 12 and x - 2y = -1. 

Hint: Consider elimination.

From x - 2y = -1, we find x = 2y - 1. Substituting into 3(2y - 1) + 4y = 12 gives 6y - 3 + 4y = 12. Simplifying gives 10y = 15, so y = 3/2. Then, x = 2(3/2) - 1 = 2. The solution is (2, 3/2).

How can you express the equation 5x + 2y = 10 in slope-intercept form?

Rearranging gives 2y = -5x + 10, or y = (-5/2)x + 5. The slope is -5/2 and the y-intercept is 5.

What does it mean if a system of equations is consistent?

A consistent system of equations has at least one solution, either one unique solution or infinitely many solutions.

What is the result of the substitution method?

The substitution method involves solving one equation for one variable and substituting that expression into the other equation to find the solution.

Solve for y in the equation: 6x + 3y = 15. 

Hint: Isolate y on one side.

Rearranging gives 3y = 15 - 6x. Dividing by 3 yields y = 5 - 2x.

Given the equations y = 2x + 3 and y = -x + 1, find the point of intersection. 

Hint: Set the equations equal to each other.

Setting 2x + 3 = -x + 1 gives 3x = -2, so x = -2/3. Substituting back, y = 2(-2/3) + 3 = 5/3. The point of intersection is (-2/3, 5/3).

What is the determinant method for solving linear equations?

The determinant method involves using Cramer's Rule, where you calculate determinants of matrices formed by the coefficients of the variables.

How can you determine the y-intercept from an equation in standard form?

To find the y-intercept from Ax + By = C, set x = 0 and solve for y.

If a linear equation is expressed as y = mx + b, what do m and b represent?

In this form, m represents the slope of the line, and b represents the y-coordinate where the line intersects the y-axis.

What does it indicate if two lines intersect at a single point?

If two lines intersect at a single point, it indicates that the system of equations has a unique solution.