Slope-intercept form

Correct answer: A. Slope -3, y-intercept 8

The governing concept is the slope-intercept form of a linear equation, y = mx + c. Here m is the slope and c is the y-intercept. Start with 12x + 4y = 32. Subtracting 12x from both sides gives 4y = -12x + 32. Dividing every term by 4 gives y = -3x + 8. Comparing the result with y = mx + c shows that m = -3 and c = 8, so option A is correct. Option B loses the negative sign when 12x is transferred to the other side. Options C and D incorrectly treat the original coefficients as the slope and intercept without first isolating y.

Correct answer: A. Slope 5/6, y-intercept 3

The governing idea is that slope and y-intercept are read after writing a linear equation as y = mx + c. Begin with 5x - 6y + 18 = 0. Moving 5x and 18 to the other side gives -6y = -5x - 18. Dividing every term by -6 changes both signs and produces y = (5/6)x + 3. Thus the coefficient of x, the slope m, is 5/6, while the constant term, the y-intercept c, is 3. Therefore option A is correct. Option B incorrectly keeps the negative signs after division by -6. Options C and D confuse the original coefficients with the values obtained in slope-intercept form, so they do not describe the line correctly.

Correct answer: A. Slope 7/8, y-intercept 3

The governing concept is slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Rearrange the equation 7x − 8y + 24 = 0 by moving the other terms: −8y = −7x − 24. Dividing every term by −8 gives y = (7/8)x + 3. Comparing this with y = mx + b, we obtain m = 7/8 and b = 3, so option A is correct. The intercept can be checked independently by putting x = 0 in the original equation: −8y + 24 = 0, which gives y = 3. Option B has both signs wrong, while options C and D use original coefficients without first isolating y.

Correct answer: A. y = 6, slope 9/10

The governing concept is the slope-intercept form y = mx + b. Rearrange 10y - 9x = 60 by adding 9x to both sides: 10y = 9x + 60. Dividing by 10 gives y = (9/10)x + 6. Therefore the slope is m = 9/10 and the y-intercept is b = 6. At x = 0, the equation gives y = (9/10)(0) + 6 = 6, so option A is correct. Option B changes signs without justification. Option C mistakes the constant 60 for the intercept, and option D reads the original coefficients directly instead of first isolating y. The positive slope also agrees with the fact that y increases as x increases.