Grades Grade 8 Expressions & EquationsSlope, Linear Equations & Systems

Slope, Linear Equations & Systems

✏️ Expressions & Equations

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Slope, Linear Equations & Systems

Slope measures steepness; systems of equations find the point where two lines intersect.

The slope of a line measures how steeply it rises or falls. In the equation y = mx + b, m is the slope and b is the y-intercept. A system of two linear equations can be solved by finding the point (x, y) that satisfies both equations simultaneously.

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

y = mx + b \quad (\text{slope-intercept form})

📌Finding slope from two points

Points: (1, 3) and (4, 9)

m = (9 − 3) / (4 − 1) = 6/3 = 2

The line rises 2 units for every 1 unit it moves right.
Equation: y = 2x + 1

m = \frac{9-3}{4-1} = \frac{6}{3} = 2 \Rightarrow y = 2x + 1

✖️Solving a system by substitution

System: y = 2x + 1 and y = −x + 7

Set equal: 2x + 1 = −x + 7
3x = 6 → x = 2
y = 2(2) + 1 = 5

Solution: (2, 5)

2x + 1 = -x + 7 \Rightarrow 3x = 6 \Rightarrow x = 2, \; y = 5

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Remember This!

Use elimination when coefficients are easy to match. Use substitution when one equation is already solved for a variable. Always verify your solution by plugging it back into BOTH original equations.

✏️ Try It!

What is the slope of the line through (−1, 4) and (3, −4)?

Practice with CalcVerse

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