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Slope equals the change in y divided by the change in x. Here the rise is 11 minus 3, which is 8, and the run is 6 minus 2, which is 4. Dividing 8 by 4 gives a slope of 2.
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting m = -3 and b = 5 gives y = -3x + 5. The coefficient of x must be the slope and the constant must be the intercept.
Add 7 to both sides to get 4x = 28, then divide both sides by 4. This gives x = 7. Checking: 4 times 7 minus 7 equals 21, which is correct.
A point lies on the line when substituting its x-value produces its y-value. For x = 1, 2 times 1 minus 1 equals 1, which matches the point (1, 1). Testing the other points shows none satisfy the equation.
A horizontal line has the same y-value at every point, so the change in y is zero. Slope equals rise over run, and zero divided by any nonzero run is 0. A vertical line, by contrast, has an undefined slope.
Distribute the 3 to get 3x + 12, then combine like terms with -2x. Since 3x minus 2x equals x, the simplified result is x + 12. Combining like terms only applies to terms with the same variable part.
A linear function has a constant rate of change, so its graph is a straight line. Parabolas, circles, and exponential curves all have changing rates of change. Only the straight line corresponds to an equation of the form y = mx + b.
A line crosses the x-axis where y equals 0. Setting -2x + 6 = 0 gives -2x = -6, so x = 3. The x-intercept is therefore the point (3, 0).
Since both expressions equal y, set them equal to each other: x + 2 = 3x - 4. Subtracting x from both sides gives 2 = 2x - 4, and adding 4 gives 6 = 2x, so x = 3. Substituting back gives y = 5, which satisfies both equations.
Use point-slope form: y - 4 = 3(x - 1). Expanding gives y - 4 = 3x - 3, and adding 4 to both sides gives y = 3x + 1. The y-intercept is 1, which can be checked by substituting x = 1 to get y = 4.
Add 5 to both sides to get 2x > 14, then divide by 2, which is positive so the inequality direction stays the same. This gives x > 7. Dividing by a negative number would be the only case that flips the sign.
Two lines are perpendicular when their slopes are negative reciprocals, meaning their product is -1. Multiplying 3/4 by -4/3 gives -1, so the lines are perpendicular. Parallel lines would need identical slopes.
The flat fee is a one-time charge that does not depend on miles, so it is the constant term. The per-mile rate multiplies the number of miles, giving C = 1.50m + 3. The slope is the rate per mile and the intercept is the flat fee.
Rewrite the equation in slope-intercept form by isolating y. Subtracting 6x gives 2y = -6x + 10, and dividing by 2 gives y = -3x + 5. The coefficient of x, -3, is the slope.
Substitute -3 for x: 4 times -3 equals -12, and -12 minus 2 equals -14. Function notation means the value inside the parentheses replaces every x in the rule.
The slope is (13 - 5)/(6 - 2) = 8/4 = 2. Using y = mx + b with the point (2, 5): 5 = 2(2) + b, so 5 = 4 + b and b = 1. The y-intercept is 1.
Parallel lines share the same slope, so the new line must have slope -5. The point (0, -3) is on the y-axis, so its y-coordinate is the y-intercept. The equation is therefore y = -5x - 3.
Distribute the 5 to get 5x - 10 = 3x + 4. Subtracting 3x from both sides gives 2x - 10 = 4, and adding 10 gives 2x = 14, so x = 7. Checking: 5 times 5 equals 25 and 3 times 7 plus 4 equals 25.
The rate of change is the change in y divided by the change in x between any two points. From x = 0 to x = 2, y increases from 4 to 10, a change of 6 over a change of 2. Dividing 6 by 2 gives 3, which is consistent across the table.
A solid boundary line means the boundary points are included, so the inequality uses ≤ or ≥. Shading below the line means y-values smaller than the boundary, so the symbol points toward y, giving y ≤ 2x + 1.
No solution means the lines never intersect, so they must be parallel. Parallel lines have equal slopes but different y-intercepts; if the intercepts were also equal, the lines would coincide and have infinitely many solutions.
Substitute x = 3 and y = 1 into the equation: 1 = 3k + 7. Subtracting 7 from both sides gives -6 = 3k, and dividing by 3 gives k = -2. Checking: -2 times 3 plus 7 equals 1.
Line A has slope (6 - 0)/(4 - 0) = 1.5. Line B is horizontal because both points share y = 8, so its slope is 0. A slope of 1.5 is greater in magnitude than 0, so line A is steeper.
Set 25h + 40 = 165, then subtract 40 to get 25h = 125. Dividing by 25 gives h = 5. Checking: 25 times 5 plus 40 equals 165.
Rewrite the second line: -2y = -4x + 6, so y = 2x - 3. For no intersection the lines must be parallel with different intercepts, so their slopes must be equal. Thus a = 2, and since the intercepts 3 and -3 differ, the lines are indeed parallel and distinct.
Subtract 7 from both sides to get 4x = 24, then divide both sides by 4 to get x = 6. Checking: 4(6) + 7 = 31, which is correct.
Distribute the 5 to get 10x - 15, then combine like terms with 4x to obtain 14x - 15. Only the x-terms combine; the constant stays as it is.
Substitute x = 2 into the rule: 3(2) - 5 = 1, which matches the y-value in the pair (2, 1). A pair is a solution only when substitution makes the equation true.
The y-intercept of a line written as y = mx + b is the constant b, which here equals 4. The y-axis is where x = 0, so the crossing point is (0, 4).
Divide both sides by -2, and because the divisor is negative the inequality sign reverses, giving x > -5. Reversing the sign is required whenever multiplying or dividing an inequality by a negative number.
A fixed monthly charge plus a per-message rate produces a linear model with constant 20 and variable term 0.10t. The total equals the flat fee added to the charge for all messages.
Substitute 4 for x to get -(4) + 9, which equals 5. The negative sign applies to the input value before adding 9.
Point-slope form is y - y1 = m(x - x1), so substituting m = -1/2 and the point (4, 1) gives y - 1 = -1/2(x - 4). The coordinates are subtracted in the same order they appear in the point.
Rewrite in slope-intercept form to get y = -2/5x + 3, so the given slope is -2/5. A perpendicular slope is the negative reciprocal, which is 5/2.
Adding the two equations eliminates y and gives 2x = 8, so x = 4. Substituting back into x + y = 7 gives y = 3.
The joining fee is paid once, so it is the constant term, while the monthly charge multiplies the number of months. This produces C = 30m + 50, a linear model with slope 30.
Two lines coincide only when they have the same slope and the same y-intercept. Since both slopes are 2, setting b equal to 5 makes the equations identical, giving infinitely many shared points.
The x-intercept occurs where y = 0, so substitute to get 3x = 12 and x = 4. The crossing point is therefore (4, 0).
The intercepts give the points (6, 0) and (0, -3). Slope is (-3 - 0)/(0 - 6) = -3/-6 = 1/2.
Distribute to get 3x + 6 = 5x - 8, then subtract 3x from both sides to get 6 = 2x - 8. Adding 8 gives 14 = 2x, so x = 7.
A dashed boundary means the line itself is excluded, so the inequality uses a strict symbol. Shading above the line corresponds to y-values greater than the line's y-values.
In a linear model written as a starting amount plus a rate times time, the coefficient of t is the constant rate of change. Because the coefficient is negative, the value drops by $1500 each year.
Parallel lines share the same slope, so line q has slope 4. Using y = 4x + b with the point (2, 3) gives 3 = 8 + b, so b = -5.
Let the smaller number be x, so the larger is 3x, and x + 3x = 24 gives 4x = 24 and x = 6. The larger number is then 3(6) = 18.
The perpendicular slope is -1/2, so the line has form y = -1/2x + b. Substituting (-2, 5) gives 5 = 1 + b, so b = 4.
For infinitely many solutions the equations must be equivalent, so the coefficients must be proportional. Since 6x is twice 3x and 12 is twice 6, ky must equal twice 4y, giving k = 4.
Subtract 3 to get -2x <= 8, then divide by -2 and reverse the sign to get x >= -4. Dividing by a negative number always flips the inequality.
Slope equals (y - 7)/(4 - 1), so (y - 7)/3 = -2. Multiplying by 3 gives y - 7 = -6, and adding 7 gives y = 1.
Break-even occurs when revenue equals cost, so set 12n = 8n + 300. Subtracting 8n gives 4n = 300, and n = 75.
A flat fee is charged once, no matter how far the ride goes, so it is the constant term. The per-mile charge is multiplied by the number of miles, so it becomes the coefficient of m. Combining these gives F = 1.50m + 4.