What is a linear equation?

A linear equation is an equation that represents a straight line when graphed on a coordinate plane. It is typically written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

What does it mean for an equation to be in slope-intercept form?

Slope-intercept form is a specific way to express linear equations, written as y = mx + b, where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.

How do you find the slope of a line given two points?

To find the slope (m) of a line given two points (x1, y1) and (x2, y2), use the formula: m = (y2 - y1) / (x2 - x1). This formula calculates the rate of change of y with respect to x.

Solve the linear equation 2x + 3 = 11. What are the steps involved?

To solve the equation 2x + 3 = 11, follow these steps: 1) Subtract 3 from both sides to isolate the term with x: 2x = 8. 2) Divide both sides by 2 to solve for x: x = 4. Thus, the solution is x = 4.

What is the significance of the y-intercept in a linear equation, and how can it be determined from the equation?

What is the formula for the point-slope form of a linear equation?

The point-slope form of a linear equation is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

If a line has a slope of 3 and passes through the point (2, 5), write its equation in slope-intercept form.

Using the point-slope form: y - 5 = 3(x - 2). Simplifying gives y = 3x - 1, which is the equation in slope-intercept form.

What is the standard form of a linear equation?

The standard form of a linear equation is written as Ax + By = C, where A, B, and C are integers, and A ≥ 0.

Find the x-intercept of the equation 4x + 2y = 8.

To find the x-intercept, set y = 0: 4x + 2(0) = 8. This simplifies to 4x = 8, so x = 2. The x-intercept is (2, 0).

If the equation of a line is y = -2x + 4, what is the slope and the y-intercept?

In the equation y = -2x + 4, the slope is -2 and the y-intercept is 4.

A linear equation has a slope of 1/2 and passes through the point (4, 6). Write the equation in point-slope form.

Using the point-slope form: y - 6 = (1/2)(x - 4).

Solve for x in the equation 5(x + 2) = 25.

Distribute 5: 5x + 10 = 25. Subtract 10 from both sides: 5x = 15. Divide by 5: x = 3.

What does it mean if two linear equations are parallel?

Two linear equations are parallel if they have the same slope (m) but different y-intercepts (b).

What is the relationship between the slopes of perpendicular lines?

The slopes of two perpendicular lines are negative reciprocals of each other, meaning m1 * m2 = -1.

Find the equation of a line that is perpendicular to y = 3x + 1 and passes through the point (1, 2).

The slope of the line y = 3x + 1 is 3, so the perpendicular slope is -1/3. Using point-slope form: y - 2 = -1/3(x - 1).

If the solution to the linear equation 3x - 4y = 12 is (x, y), what is the value of y when x = 0?

Substituting x = 0 in the equation: 3(0) - 4y = 12 gives -4y = 12, so y = -3.