This cheat sheet covers the three main forms of linear equations: slope-intercept form, point-slope form, and standard form. Students need these forms to graph lines, write equations from given information, and compare linear relationships. Each form shows different information quickly, so choosing the right one can make a problem much easier.
The sheet is designed as a clear reference for classwork, homework, and test review.
Slope-intercept form, , shows the slope and -intercept directly. Point-slope form, , is useful when you know a point and the slope. Standard form, , is useful for finding intercepts and comparing equations.
Converting between forms helps students graph accurately and understand that the same line can be written in different ways.
Key Facts
- Slope-intercept form is , where is the slope and is the -intercept.
- Point-slope form is , where is a point on the line and is the slope.
- Standard form is , where , , and are usually integers and is often written as nonnegative.
- Slope is calculated by when two points and are known.
- To graph , start at the -intercept and use the slope .
- To find the -intercept from , set and solve .
- To find the -intercept from , set and solve .
- Horizontal lines have slope and equations like , while vertical lines have undefined slope and equations like .
Vocabulary
- Slope
- Slope is the rate of change of a line, found by comparing vertical change to horizontal change.
- Y-intercept
- The -intercept is the point where a line crosses the -axis, written as .
- X-intercept
- The -intercept is the point where a line crosses the -axis, found by setting .
- Slope-intercept form
- Slope-intercept form is , which directly shows the slope and the -intercept.
- Point-slope form
- Point-slope form is , which uses one point on a line and the slope.
- Standard form
- Standard form is , a linear equation form often used to find intercepts.
Common Mistakes to Avoid
- Confusing slope and -intercept in is wrong because gives the rate of change and gives the starting point on the -axis.
- Using is wrong because slope must be vertical change over horizontal change, .
- Forgetting the subtraction signs in is wrong because the signs depend on the coordinates of the known point.
- Treating as slope-intercept form is wrong because the slope and -intercept are not visible until the equation is solved for .
- Calling a vertical line's slope is wrong because horizontal lines have slope , while vertical lines have undefined slope.
Practice Questions
- 1 Write the equation of a line in slope-intercept form with slope and -intercept .
- 2 Find the slope of the line through and using .
- 3 Convert to slope-intercept form and identify the slope and -intercept.
- 4 Explain which form of a linear equation is most useful when you know a point on the line and the slope, and explain why.
Understanding Slope Forms (Slope-Intercept, Point-Slope, Standard)
A line represents a constant rate of change. This means that every equal step across the graph produces the same vertical change. A taxi fare can have a starting charge followed by a fixed cost per mile.
A savings account can begin with some money and grow by the same deposit each week. In these situations, the starting amount is where the line crosses the vertical axis. The rate tells how quickly the output changes as the input changes.
A positive rate rises from left to right. A negative rate falls.
The size of the rate matters too. A line with a rate of three changes three vertical units for each one horizontal unit, so it is steeper than a line with a rate of one half.
When finding a rate from two points, keep the order of the points consistent. Subtract the second vertical value from the first vertical value, then subtract the second horizontal value from the first horizontal value in the same order. Reversing both subtractions gives the same result.
Reversing only one gives the wrong sign. Fractions are useful here because a rate can describe many equivalent moves. Up two and right three has the same rate as up four and right six.
On a graph, choose moves that land on clear grid intersections. This reduces counting errors.
A rate with zero on top makes a horizontal line. A vertical line cannot have a rate because its horizontal change is zero, and division by zero is not defined.
Changing equation forms is mostly careful algebra. To turn a point and rate equation into a form that reveals the vertical intercept, distribute the rate to every term inside the parentheses. Then combine number terms and isolate the vertical variable.
A common error is forgetting that a negative rate changes the sign of each term it multiplies. To turn an equation with both variables on one side into slope form, move the horizontal term away from the vertical term, then divide every term by the coefficient of the vertical variable. This division can change signs.
Check the result by substituting a known point. If both sides have the same value, the equation still describes that point.
Standard form is especially helpful when a problem focuses on boundaries, totals, or intercepts. For example, a school fundraiser may track the number of two ticket types sold when each type has a different price and the total money is fixed. The intercepts show what happens if only one ticket type is sold.
An intercept is meaningful only when zero is sensible in the situation. Negative ticket sales do not make sense, even though a graph may extend into negative values. When comparing two lines, equal rates mean the lines are parallel.
Equal rates with equal starting values mean they are the same line. Different rates meet at one point, and that meeting point often represents a real balance, such as two payment plans costing the same amount.